data/curriculum.json

{
  "pathways": [
    {
      "id": "precalculus",
      "title": "Pre-Calculus Fundamentals",
      "level": "Foundational",
      "description": "Build the essential math foundation needed before calculus. Master functions, trigonometry, exponentials, sequences, and more so you are fully prepared for Calculus I.",
      "chapters": [
        {
          "id": "precalc_ch1_functions_graphs",
          "title": "Functions and Graphs",
          "description": "Learn what functions are, how to read and build their graphs, and how to work with all the major function families you will encounter throughout calculus.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch1_s01",
              "type": "lesson",
              "title": "What Is a Function?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A function is a rule that takes an input and produces exactly one output. Think of it like a machine: you feed a number in, the machine does something to it, and one number comes out. If you put the same number in twice, you always get the same result."
                },
                {
                  "kind": "text",
                  "text": "We write functions using notation like f(x) = 2x + 3. Here, f is the name of the function and x is the input. If you plug in x = 4, you get f(4) = 2(4) + 3 = 11. The key rule is: every input gives exactly one output. A single input can never produce two different outputs."
                },
                {
                  "kind": "example",
                  "text": "Suppose f(x) = x^2. Then f(3) = 9, f(-3) = 9, and f(0) = 0. Notice that different inputs can give the same output (3 and -3 both give 9), but each input only gives one output."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "function_machine_diagram"
                }
              ],
              "related_topic_ids": [
                "topic_functions_intro"
              ],
              "glossary_terms": [
                "function",
                "input",
                "output",
                "vertical line test"
              ]
            },
            {
              "id": "precalc_ch1_s02",
              "type": "lesson",
              "title": "What Is a Function?",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "The vertical line test is a quick way to check if a graph represents a function. If any vertical line crosses the graph more than once, it is not a function because that x-value would have two outputs."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_functions_intro"
              ],
              "glossary_terms": [
                "function",
                "input",
                "output",
                "vertical line test"
              ]
            },
            {
              "id": "precalc_ch1_s03",
              "type": "lesson",
              "title": "Function Notation and Evaluation",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Function notation is the standard way we write and talk about functions. When we write f(x) = 3x - 1, we are saying: the function named f takes input x and returns 3x - 1. The letter f is just a name \u2014 we could also call it g, h, or anything else."
                },
                {
                  "kind": "text",
                  "text": "Evaluating a function means substituting a specific value for x and computing the result. For f(x) = 3x - 1, evaluating at x = 5 means computing f(5) = 3(5) - 1 = 14. You can also evaluate at expressions: f(a + 1) = 3(a + 1) - 1 = 3a + 3 - 1 = 3a + 2."
                },
                {
                  "kind": "example",
                  "text": "Let g(x) = x^2 - 4x + 7. Find g(2). Substitute 2 for every x: g(2) = (2)^2 - 4(2) + 7 = 4 - 8 + 7 = 3."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "function_notation_breakdown"
                }
              ],
              "related_topic_ids": [
                "topic_functions_intro",
                "topic_function_notation"
              ],
              "glossary_terms": [
                "function",
                "evaluate",
                "function notation"
              ]
            },
            {
              "id": "precalc_ch1_s04",
              "type": "lesson",
              "title": "Function Notation and Evaluation",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "Be careful with negative inputs. For f(x) = x^2, computing f(-3) means (-3)^2 = 9, not -(3^2) = -9. Always wrap the substituted value in parentheses to avoid sign errors."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_functions_intro",
                "topic_function_notation"
              ],
              "glossary_terms": [
                "function",
                "evaluate",
                "function notation"
              ]
            },
            {
              "id": "precalc_ch1_s05",
              "type": "lesson",
              "title": "Domain and Range",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The domain of a function is the set of all inputs (x-values) the function can accept. The range is the set of all outputs (y-values) the function can produce. Together, domain and range tell you the full scope of what a function can do."
                },
                {
                  "kind": "text",
                  "text": "Some functions accept any real number as input. For example, f(x) = 2x + 1 works for every x, so its domain is all real numbers. Other functions have restrictions. For f(x) = 1/x, you cannot use x = 0 because division by zero is undefined. So the domain is all real numbers except 0."
                },
                {
                  "kind": "text",
                  "text": "For f(x) = sqrt(x), the input must be zero or positive because you cannot take the square root of a negative number (in real numbers). So the domain is x >= 0. The range of sqrt(x) is y >= 0 because square roots are never negative."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "domain_range_visual_sqrt"
                }
              ],
              "related_topic_ids": [
                "topic_domain_range"
              ],
              "glossary_terms": [
                "domain",
                "range",
                "square root",
                "undefined"
              ]
            },
            {
              "id": "precalc_ch1_s06",
              "type": "lesson",
              "title": "Domain and Range",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Find the domain of f(x) = 1/(x - 3). The denominator cannot be zero, so x - 3 \u2260 0, meaning x \u2260 3. The domain is all real numbers except x = 3."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_domain_range"
              ],
              "glossary_terms": [
                "domain",
                "range",
                "square root",
                "undefined"
              ]
            },
            {
              "id": "precalc_ch1_s07",
              "type": "lesson",
              "title": "Linear Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its graph is always a straight line. Linear functions are the simplest type of function and serve as a building block for understanding all others."
                },
                {
                  "kind": "text",
                  "text": "The slope m tells you how steep the line is and which direction it tilts. A positive slope means the line rises from left to right. A negative slope means it falls. A slope of zero gives a perfectly horizontal line. The y-intercept b is where the line crosses the y-axis (the point (0, b))."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "linear_function_slope_intercept"
                }
              ],
              "related_topic_ids": [
                "topic_linear_functions"
              ],
              "glossary_terms": [
                "linear function",
                "slope",
                "y-intercept",
                "rate of change"
              ]
            },
            {
              "id": "precalc_ch1_s08",
              "type": "lesson",
              "title": "Linear Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Slope is calculated as the change in y divided by the change in x between any two points: m = (y2 - y1)/(x2 - x1). This ratio is constant everywhere on a straight line, which is exactly what makes the function linear \u2014 it changes at a constant rate."
                },
                {
                  "kind": "example",
                  "text": "The line through (1, 2) and (4, 8) has slope m = (8 - 2)/(4 - 1) = 6/3 = 2. Using point-slope form: y - 2 = 2(x - 1), which simplifies to y = 2x. The y-intercept is 0."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_linear_functions"
              ],
              "glossary_terms": [
                "linear function",
                "slope",
                "y-intercept",
                "rate of change"
              ]
            },
            {
              "id": "precalc_ch1_s09",
              "type": "worked_example",
              "title": "Worked Example: Analyzing a Linear Function",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "A phone plan charges a $20 monthly fee plus $0.05 per text message. Write a function for the monthly cost C(t) where t is the number of texts. Find the cost for 200 texts. What is the slope and what does it mean?"
                },
                {
                  "kind": "step",
                  "text": "Step 1: Identify the structure. The cost has a fixed part ($20) plus a variable part ($0.05 per text). This is linear: C(t) = 0.05t + 20."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Evaluate at t = 200. C(200) = 0.05(200) + 20 = 10 + 20 = 30. The monthly cost for 200 texts is $30."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "phone_plan_linear_graph"
                }
              ],
              "related_topic_ids": [
                "topic_linear_functions",
                "topic_function_notation"
              ],
              "glossary_terms": [
                "linear function",
                "slope",
                "y-intercept",
                "rate of change"
              ]
            },
            {
              "id": "precalc_ch1_s10",
              "type": "worked_example",
              "title": "Worked Example: Analyzing a Linear Function",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 3: Identify the slope. The slope is 0.05. It means each additional text message adds $0.05 to the bill. The slope is the rate of change of cost with respect to the number of texts."
                },
                {
                  "kind": "step",
                  "text": "Step 4: Identify the y-intercept. The y-intercept is 20. It represents the base cost when t = 0 (no texts sent). You still pay $20 just for having the plan."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_linear_functions",
                "topic_function_notation"
              ],
              "glossary_terms": [
                "linear function",
                "slope",
                "y-intercept",
                "rate of change"
              ]
            },
            {
              "id": "precalc_ch1_s11",
              "type": "lesson",
              "title": "Quadratic Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A quadratic function has the form f(x) = ax^2 + bx + c, where a \u2260 0. Its graph is a U-shaped curve called a parabola. If a is positive, the parabola opens upward (like a bowl). If a is negative, it opens downward (like an upside-down bowl)."
                },
                {
                  "kind": "text",
                  "text": "The most important point on a parabola is the vertex \u2014 the peak or valley of the curve. The vertex is located at x = -b/(2a). Plugging this x back into f(x) gives you the y-coordinate of the vertex. The vertical line through the vertex is the axis of symmetry: the parabola is a mirror image on either side."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "quadratic_parabola_vertex"
                }
              ],
              "related_topic_ids": [
                "topic_quadratic_functions"
              ],
              "glossary_terms": [
                "quadratic function",
                "parabola",
                "vertex",
                "axis of symmetry",
                "discriminant",
                "zeros"
              ]
            },
            {
              "id": "precalc_ch1_s12",
              "type": "lesson",
              "title": "Quadratic Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The zeros (or roots) of a quadratic are the x-values where f(x) = 0. You can find them by factoring, completing the square, or using the quadratic formula: x = (-b \u00b1 sqrt(b^2 - 4ac))/(2a). The expression b^2 - 4ac is called the discriminant. If it is positive, there are two real zeros. If zero, there is one. If negative, there are no real zeros."
                },
                {
                  "kind": "example",
                  "text": "For f(x) = x^2 - 6x + 8, the vertex is at x = -(-6)/(2\u00b71) = 3, giving f(3) = 9 - 18 + 8 = -1. The vertex is (3, -1). Setting f(x) = 0: x^2 - 6x + 8 = (x-2)(x-4) = 0, so zeros are x = 2 and x = 4."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_quadratic_functions"
              ],
              "glossary_terms": [
                "quadratic function",
                "parabola",
                "vertex",
                "axis of symmetry",
                "discriminant",
                "zeros"
              ]
            },
            {
              "id": "precalc_ch1_s13",
              "type": "lesson",
              "title": "Polynomial Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A polynomial function is a sum of terms where each term is a constant times x raised to a whole-number power. Examples include f(x) = x^3 - 2x + 5 and g(x) = 4x^5 - x^2 + 3. The highest power of x is called the degree of the polynomial. Linear functions have degree 1, quadratics have degree 2, cubics have degree 3, and so on."
                },
                {
                  "kind": "text",
                  "text": "The degree tells you a lot about the graph's behavior. A polynomial of degree n can have at most n real zeros (x-intercepts) and at most n - 1 turning points (peaks and valleys). The end behavior \u2014 what happens as x goes to positive or negative infinity \u2014 depends on the leading term (the highest-degree term)."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "polynomial_end_behavior"
                }
              ],
              "related_topic_ids": [
                "topic_polynomial_functions"
              ],
              "glossary_terms": [
                "polynomial",
                "degree",
                "zeros",
                "end behavior",
                "leading term",
                "turning point"
              ]
            },
            {
              "id": "precalc_ch1_s14",
              "type": "lesson",
              "title": "Polynomial Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "For example, if the leading term is x^3, then as x goes to positive infinity, f(x) also goes to positive infinity, and as x goes to negative infinity, f(x) goes to negative infinity. If the leading term is -x^4, the graph falls on both ends because the negative sign flips the curve and even-degree polynomials behave the same on both sides."
                },
                {
                  "kind": "example",
                  "text": "f(x) = x^3 - 3x has degree 3. Its zeros: x(x^2 - 3) = 0 gives x = 0, x = sqrt(3), x = -sqrt(3). It has at most 2 turning points. End behavior: falls to the left, rises to the right."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_polynomial_functions"
              ],
              "glossary_terms": [
                "polynomial",
                "degree",
                "zeros",
                "end behavior",
                "leading term",
                "turning point"
              ]
            },
            {
              "id": "precalc_ch1_s15",
              "type": "lesson",
              "title": "Rational Functions and Asymptotes",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A rational function is a ratio of two polynomials: f(x) = P(x)/Q(x). For example, f(x) = (x + 1)/(x - 2). These functions behave differently from polynomials because the denominator can equal zero, creating gaps or blow-ups in the graph."
                },
                {
                  "kind": "text",
                  "text": "A vertical asymptote occurs at any x-value that makes the denominator zero (and does not cancel with the numerator). The graph shoots up or down toward infinity near these points. For f(x) = (x+1)/(x-2), there is a vertical asymptote at x = 2."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "rational_function_asymptotes"
                }
              ],
              "related_topic_ids": [
                "topic_rational_functions"
              ],
              "glossary_terms": [
                "rational function",
                "vertical asymptote",
                "horizontal asymptote",
                "asymptote"
              ]
            },
            {
              "id": "precalc_ch1_s16",
              "type": "lesson",
              "title": "Rational Functions and Asymptotes",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A horizontal asymptote describes what happens to f(x) as x gets very large or very small. If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is the ratio of their leading coefficients. If the numerator degree is less, the horizontal asymptote is y = 0. If the numerator degree is greater, there is no horizontal asymptote (but there may be an oblique one)."
                },
                {
                  "kind": "example",
                  "text": "For f(x) = (2x + 1)/(x - 3): vertical asymptote at x = 3. Both numerator and denominator have degree 1, so horizontal asymptote is y = 2/1 = 2. As x gets huge, f(x) approaches 2."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_rational_functions"
              ],
              "glossary_terms": [
                "rational function",
                "vertical asymptote",
                "horizontal asymptote",
                "asymptote"
              ]
            },
            {
              "id": "precalc_ch1_s17",
              "type": "worked_example",
              "title": "Worked Example: Analyzing a Rational Function",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Find the domain, vertical asymptote(s), and horizontal asymptote of f(x) = (3x)/(x^2 - 4). Describe what happens near each asymptote."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Find the domain. Set the denominator equal to zero: x^2 - 4 = 0, so (x-2)(x+2) = 0, giving x = 2 and x = -2. The domain is all real numbers except x = 2 and x = -2."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Check for vertical asymptotes. The numerator 3x does not equal zero at x = 2 or x = -2 (it equals 6 and -6 respectively), so nothing cancels. There are vertical asymptotes at x = 2 and x = -2."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "rational_analysis_3x_over_x2minus4"
                }
              ],
              "related_topic_ids": [
                "topic_rational_functions",
                "topic_domain_range"
              ],
              "glossary_terms": [
                "rational function",
                "domain",
                "vertical asymptote",
                "horizontal asymptote"
              ]
            },
            {
              "id": "precalc_ch1_s18",
              "type": "worked_example",
              "title": "Worked Example: Analyzing a Rational Function",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 3: Find the horizontal asymptote. The numerator has degree 1 and the denominator has degree 2. Since the denominator degree is larger, the horizontal asymptote is y = 0. As x gets very large, f(x) approaches 0."
                },
                {
                  "kind": "step",
                  "text": "Step 4: Describe behavior near asymptotes. Near x = 2, the function blows up to positive infinity on one side and negative infinity on the other. The same happens near x = -2. The graph gets closer and closer to y = 0 as you move far left or far right."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_rational_functions",
                "topic_domain_range"
              ],
              "glossary_terms": [
                "rational function",
                "domain",
                "vertical asymptote",
                "horizontal asymptote"
              ]
            },
            {
              "id": "precalc_ch1_s19",
              "type": "lesson",
              "title": "Graph Transformations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Once you know the graph of a basic function, you can create many new graphs by applying transformations. There are four main types: vertical shifts, horizontal shifts, vertical stretches or compressions, and reflections."
                },
                {
                  "kind": "text",
                  "text": "Vertical shift: f(x) + k moves the graph up by k units (down if k is negative). Horizontal shift: f(x - h) moves the graph right by h units (left if h is negative). Notice the sign is opposite \u2014 f(x - 3) shifts right, not left."
                },
                {
                  "kind": "text",
                  "text": "Vertical stretch/compression: a\u00b7f(x) stretches the graph vertically by factor a if |a| > 1, or compresses it if |a| < 1. If a is negative, the graph also flips over the x-axis. Horizontal stretch/compression: f(bx) compresses horizontally by factor b if |b| > 1."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "graph_transformations_four_types"
                }
              ],
              "related_topic_ids": [
                "topic_graph_transformations"
              ],
              "glossary_terms": [
                "transformation",
                "vertical shift",
                "horizontal shift",
                "reflection",
                "stretch",
                "compression"
              ]
            },
            {
              "id": "precalc_ch1_s20",
              "type": "lesson",
              "title": "Graph Transformations",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Start with f(x) = x^2. Then g(x) = (x - 2)^2 + 3 is the same parabola shifted 2 units right and 3 units up. The vertex moves from (0, 0) to (2, 3). And h(x) = -x^2 is the parabola flipped upside-down."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_graph_transformations"
              ],
              "glossary_terms": [
                "transformation",
                "vertical shift",
                "horizontal shift",
                "reflection",
                "stretch",
                "compression"
              ]
            },
            {
              "id": "precalc_ch1_s21",
              "type": "lesson",
              "title": "Function Composition and Inverses",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Function composition means using the output of one function as the input of another. If f(x) = 2x and g(x) = x + 3, then the composition (f \u2218 g)(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6. You evaluate the inner function first, then feed its result into the outer function."
                },
                {
                  "kind": "text",
                  "text": "The order matters: (f \u2218 g)(x) is usually different from (g \u2218 f)(x). Using the same functions: (g \u2218 f)(x) = g(f(x)) = g(2x) = 2x + 3, which is not the same as 2x + 6."
                },
                {
                  "kind": "text",
                  "text": "An inverse function undoes what the original function does. If f(x) = 2x + 3, its inverse f^(-1)(x) reverses the process: subtract 3, then divide by 2, giving f^(-1)(x) = (x - 3)/2. Composing a function with its inverse always returns the original input: f(f^(-1)(x)) = x."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "composition_and_inverse_diagram"
                }
              ],
              "related_topic_ids": [
                "topic_function_composition",
                "topic_inverse_functions"
              ],
              "glossary_terms": [
                "composition",
                "inverse function",
                "one-to-one",
                "horizontal line test"
              ]
            },
            {
              "id": "precalc_ch1_s22",
              "type": "lesson",
              "title": "Function Composition and Inverses",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "Not every function has an inverse. A function must be one-to-one (each output comes from exactly one input) to have an inverse. The horizontal line test checks this: if any horizontal line crosses the graph more than once, the function is not one-to-one."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_function_composition",
                "topic_inverse_functions"
              ],
              "glossary_terms": [
                "composition",
                "inverse function",
                "one-to-one",
                "horizontal line test"
              ]
            },
            {
              "id": "precalc_ch1_s23",
              "type": "practice",
              "title": "Practice: Functions and Graphs",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) For f(x) = (x - 1)/(x + 4), find the domain, the vertical asymptote, and the horizontal asymptote."
                },
                {
                  "kind": "problem",
                  "text": "2) Let g(x) = x^2 + 2x - 8. Find the vertex, the axis of symmetry, and the zeros of g."
                },
                {
                  "kind": "problem",
                  "text": "3) Given f(x) = 3x - 5 and g(x) = x^2, compute (f \u2218 g)(2) and (g \u2218 f)(2)."
                },
                {
                  "kind": "problem",
                  "text": "4) Describe the transformations applied to f(x) = |x| to obtain g(x) = -2|x - 1| + 4."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_functions_intro",
                "topic_domain_range",
                "topic_quadratic_functions",
                "topic_function_composition"
              ],
              "glossary_terms": [
                "domain",
                "vertex",
                "composition",
                "transformation"
              ]
            },
            {
              "id": "precalc_ch1_s24",
              "type": "summary",
              "title": "Chapter Summary: Functions and Graphs",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A function assigns exactly one output to each input. We use f(x) notation to name and evaluate functions. Every function has a domain (allowed inputs) and range (possible outputs)."
                },
                {
                  "kind": "text",
                  "text": "Linear functions (y = mx + b) have constant rate of change and straight-line graphs. Quadratic functions (y = ax^2 + bx + c) form parabolas with a vertex and axis of symmetry. Polynomials generalize these with higher degrees, and their end behavior is controlled by the leading term."
                },
                {
                  "kind": "text",
                  "text": "Rational functions (ratios of polynomials) introduce asymptotes \u2014 lines the graph approaches but never crosses. Vertical asymptotes occur where the denominator is zero; horizontal asymptotes describe long-run behavior."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "functions_chapter_summary_map"
                }
              ],
              "related_topic_ids": [
                "topic_functions_intro",
                "topic_linear_functions",
                "topic_quadratic_functions",
                "topic_polynomial_functions",
                "topic_rational_functions"
              ],
              "glossary_terms": [
                "function",
                "domain",
                "range",
                "slope",
                "parabola",
                "asymptote",
                "transformation",
                "composition",
                "inverse function"
              ]
            },
            {
              "id": "precalc_ch1_s25",
              "type": "summary",
              "title": "Chapter Summary: Functions and Graphs",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Transformations (shifts, stretches, reflections) let you build new graphs from familiar ones. Composition chains functions together, and inverse functions undo each other. These core ideas will appear constantly in calculus."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_functions_intro",
                "topic_linear_functions",
                "topic_quadratic_functions",
                "topic_polynomial_functions",
                "topic_rational_functions"
              ],
              "glossary_terms": [
                "function",
                "domain",
                "range",
                "slope",
                "parabola",
                "asymptote",
                "transformation",
                "composition",
                "inverse function"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch1_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Functions and Graphs",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch1_test",
            "optional_recommended": true,
            "title": "Chapter Test: Functions and Graphs",
            "questions": []
          }
        },
        {
          "id": "precalc_ch2_exp_log",
          "title": "Exponential and Logarithmic Functions",
          "description": "Understand exponential growth and decay, learn what logarithms are and why they matter, and practice solving equations involving these powerful function families.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch2_s01",
              "type": "lesson",
              "title": "What Is an Exponential Function?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "An exponential function has the form f(x) = a \u00b7 b^x, where a is the starting value and b is the base. The base b must be positive and not equal to 1. What makes exponentials special is that the variable x is in the exponent, not the base."
                },
                {
                  "kind": "text",
                  "text": "This creates a distinctive growth pattern. Instead of adding a fixed amount each step (like linear functions), exponential functions multiply by a fixed factor each step. If b > 1, the function grows \u2014 slowly at first, then faster and faster. If 0 < b < 1, the function decays, shrinking toward zero."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "exponential_growth_decay_comparison"
                }
              ],
              "related_topic_ids": [
                "topic_exponential_functions"
              ],
              "glossary_terms": [
                "exponential function",
                "base",
                "exponent",
                "horizontal asymptote"
              ]
            },
            {
              "id": "precalc_ch2_s02",
              "type": "lesson",
              "title": "What Is an Exponential Function?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The graph of an exponential function always passes through (0, a) because b^0 = 1 for any base b, so f(0) = a \u00b7 1 = a. The graph has a horizontal asymptote at y = 0: the function gets close to zero but never actually reaches it (for decay) or rises from near zero (for growth)."
                },
                {
                  "kind": "example",
                  "text": "f(x) = 3 \u00b7 2^x starts at f(0) = 3. At x = 1 it is 6, at x = 2 it is 12, at x = 3 it is 24. Each step doubles the previous value. That is the power of exponential growth."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions"
              ],
              "glossary_terms": [
                "exponential function",
                "base",
                "exponent",
                "horizontal asymptote"
              ]
            },
            {
              "id": "precalc_ch2_s03",
              "type": "lesson",
              "title": "Growth, Decay, and Real-World Models",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Exponential functions model countless real-world situations. When a quantity increases by a fixed percentage each time period, it grows exponentially. When it decreases by a fixed percentage, it decays exponentially."
                },
                {
                  "kind": "text",
                  "text": "Population growth: if a town of 10,000 grows by 3% per year, the population after t years is P(t) = 10000 \u00b7 (1.03)^t. Radioactive decay: if a substance loses half its mass every 5 years (half-life = 5), the remaining mass after t years is M(t) = M_0 \u00b7 (0.5)^(t/5)."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "population_growth_model"
                }
              ],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_exponential_models"
              ],
              "glossary_terms": [
                "exponential growth",
                "exponential decay",
                "half-life",
                "compound interest"
              ]
            },
            {
              "id": "precalc_ch2_s04",
              "type": "lesson",
              "title": "Growth, Decay, and Real-World Models",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Compound interest is another classic example. If you invest P dollars at an annual rate r compounded n times per year, after t years you have A = P \u00b7 (1 + r/n)^(nt). As n gets larger and larger (compounding more frequently), this approaches A = P \u00b7 e^(rt), which uses continuous compounding."
                },
                {
                  "kind": "note",
                  "text": "Exponential growth is deceptively fast. A population growing at just 2% per year will double in about 35 years. Understanding this is important for biology, finance, and many other fields."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_exponential_models"
              ],
              "glossary_terms": [
                "exponential growth",
                "exponential decay",
                "half-life",
                "compound interest"
              ]
            },
            {
              "id": "precalc_ch2_s05",
              "type": "lesson",
              "title": "The Special Number e",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Among all possible bases for exponential functions, one stands out: the number e, approximately 2.71828. It is called Euler's number and appears naturally whenever growth or decay happens continuously."
                },
                {
                  "kind": "text",
                  "text": "Where does e come from? Imagine investing $1 at 100% annual interest. Compounded once: you get $2. Compounded twice (50% each half-year): $2.25. Compounded monthly: about $2.61. Compounded every second: about $2.71828. As you compound more and more frequently, the result approaches e. Formally, e = limit of (1 + 1/n)^n as n approaches infinity."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "natural_exponential_e_x"
                }
              ],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_number_e"
              ],
              "glossary_terms": [
                "Euler's number",
                "natural exponential",
                "continuous growth"
              ]
            },
            {
              "id": "precalc_ch2_s06",
              "type": "lesson",
              "title": "The Special Number e",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The function f(x) = e^x is called the natural exponential function. It has a remarkable property that will become central in calculus: its rate of change at any point equals its value at that point. This makes e the most natural base for modeling continuous processes."
                },
                {
                  "kind": "example",
                  "text": "Continuous growth model: a colony of bacteria doubles every hour. Using the continuous model, the count is N(t) = N_0 \u00b7 e^(kt), where k = ln(2) \u2248 0.693. So if N_0 = 100, after 3 hours: N(3) = 100 \u00b7 e^(0.693\u00b73) \u2248 100 \u00b7 8 = 800."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_number_e"
              ],
              "glossary_terms": [
                "Euler's number",
                "natural exponential",
                "continuous growth"
              ]
            },
            {
              "id": "precalc_ch2_s07",
              "type": "worked_example",
              "title": "Worked Example: Modeling with Exponentials",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "A car worth $25,000 depreciates by 15% each year. Write a function for its value V(t) after t years. What is it worth after 4 years? When will it be worth less than $10,000?"
                },
                {
                  "kind": "step",
                  "text": "Step 1: Set up the model. Each year the car keeps 85% of its value (100% - 15% = 85%). So V(t) = 25000 \u00b7 (0.85)^t."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Find V(4). V(4) = 25000 \u00b7 (0.85)^4 = 25000 \u00b7 0.52200625 \u2248 $13,050.16. After 4 years the car is worth about $13,050."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "car_depreciation_graph"
                }
              ],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_exponential_models"
              ],
              "glossary_terms": [
                "exponential decay",
                "depreciation"
              ]
            },
            {
              "id": "precalc_ch2_s08",
              "type": "worked_example",
              "title": "Worked Example: Modeling with Exponentials",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 3: Solve 25000 \u00b7 (0.85)^t < 10000. Divide both sides by 25000: (0.85)^t < 0.4. Take the natural log of both sides: t \u00b7 ln(0.85) < ln(0.4). Since ln(0.85) is negative, dividing flips the inequality: t > ln(0.4)/ln(0.85) \u2248 (-0.9163)/(-0.1625) \u2248 5.64."
                },
                {
                  "kind": "step",
                  "text": "Step 4: Interpret. The car drops below $10,000 after about 5.64 years, so partway through the 6th year."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_exponential_models"
              ],
              "glossary_terms": [
                "exponential decay",
                "depreciation"
              ]
            },
            {
              "id": "precalc_ch2_s09",
              "type": "lesson",
              "title": "What Is a Logarithm?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A logarithm answers the question: what exponent do I need? Specifically, log_b(x) = y means b^y = x. In words, log base b of x is the power you raise b to in order to get x."
                },
                {
                  "kind": "text",
                  "text": "For example, log_2(8) = 3 because 2^3 = 8. And log_10(1000) = 3 because 10^3 = 1000. Logarithms are the inverse of exponential functions \u2014 they undo exponentiation, just like subtraction undoes addition."
                },
                {
                  "kind": "text",
                  "text": "The graph of y = log_b(x) is the reflection of y = b^x over the line y = x. It passes through (1, 0) because b^0 = 1 for any base. The domain is x > 0 (you can only take the log of a positive number) and the range is all real numbers. There is a vertical asymptote at x = 0."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "logarithm_graph_base2"
                }
              ],
              "related_topic_ids": [
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "logarithm",
                "base",
                "inverse function"
              ]
            },
            {
              "id": "precalc_ch2_s10",
              "type": "lesson",
              "title": "What Is a Logarithm?",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "log_5(25) = 2 because 5^2 = 25. log_3(1/9) = -2 because 3^(-2) = 1/9. log_10(1) = 0 because 10^0 = 1."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "logarithm",
                "base",
                "inverse function"
              ]
            },
            {
              "id": "precalc_ch2_s11",
              "type": "lesson",
              "title": "Properties of Logarithms",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Logarithms have three key properties that come directly from the rules of exponents. These properties turn multiplication into addition, division into subtraction, and powers into multiplication \u2014 which is extremely useful for simplifying and solving equations."
                },
                {
                  "kind": "text",
                  "text": "Product rule: log_b(A \u00b7 B) = log_b(A) + log_b(B). The log of a product is the sum of the logs. Quotient rule: log_b(A/B) = log_b(A) - log_b(B). The log of a quotient is the difference of the logs. Power rule: log_b(A^n) = n \u00b7 log_b(A). The log of a power brings the exponent down as a multiplier."
                },
                {
                  "kind": "text",
                  "text": "These properties work because exponents add when you multiply same-base powers: b^m \u00b7 b^n = b^(m+n). Since logs are exponents, the same addition relationship holds."
                }
              ],
              "graphics": [
                {
                  "kind": "table",
                  "name": "log_properties_summary_table"
                }
              ],
              "related_topic_ids": [
                "topic_logarithmic_functions",
                "topic_log_properties"
              ],
              "glossary_terms": [
                "logarithm",
                "product rule",
                "quotient rule",
                "power rule"
              ]
            },
            {
              "id": "precalc_ch2_s12",
              "type": "lesson",
              "title": "Properties of Logarithms",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Expand log_2(8x^3). Using the product rule: log_2(8) + log_2(x^3). Then the power rule: 3 + 3\u00b7log_2(x). So log_2(8x^3) = 3 + 3\u00b7log_2(x)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_logarithmic_functions",
                "topic_log_properties"
              ],
              "glossary_terms": [
                "logarithm",
                "product rule",
                "quotient rule",
                "power rule"
              ]
            },
            {
              "id": "precalc_ch2_s13",
              "type": "lesson",
              "title": "Common and Natural Logarithms",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Two bases are used so often they get special names and shorthand. The common logarithm uses base 10 and is written simply as log(x) (no base written). The natural logarithm uses base e and is written as ln(x)."
                },
                {
                  "kind": "text",
                  "text": "Common logs are handy for everyday scales. The Richter scale for earthquakes, the decibel scale for sound, and pH in chemistry all use log base 10. Going up by 1 on these scales means multiplying the actual quantity by 10."
                },
                {
                  "kind": "text",
                  "text": "Natural logs appear throughout calculus and science because they pair with the natural exponential e^x. The key relationships are: ln(e^x) = x and e^(ln x) = x. These show that ln and e^x are perfect inverses of each other."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "common_vs_natural_log_graph"
                }
              ],
              "related_topic_ids": [
                "topic_logarithmic_functions",
                "topic_number_e"
              ],
              "glossary_terms": [
                "common logarithm",
                "natural logarithm",
                "change of base formula"
              ]
            },
            {
              "id": "precalc_ch2_s14",
              "type": "lesson",
              "title": "Common and Natural Logarithms",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The change of base formula lets you convert between any two bases: log_b(x) = ln(x)/ln(b) = log(x)/log(b). This is useful when your calculator only has log and ln buttons."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_logarithmic_functions",
                "topic_number_e"
              ],
              "glossary_terms": [
                "common logarithm",
                "natural logarithm",
                "change of base formula"
              ]
            },
            {
              "id": "precalc_ch2_s15",
              "type": "worked_example",
              "title": "Worked Example: Solving a Logarithmic Equation",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Solve for x: log_3(2x + 1) = 4."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Convert from logarithmic to exponential form. log_3(2x + 1) = 4 means 3^4 = 2x + 1."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Compute 3^4 = 81. So the equation becomes 81 = 2x + 1."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Solve for x. Subtract 1: 80 = 2x. Divide by 2: x = 40."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "logarithm",
                "exponential form"
              ]
            },
            {
              "id": "precalc_ch2_s16",
              "type": "worked_example",
              "title": "Worked Example: Solving a Logarithmic Equation",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Check. log_3(2(40) + 1) = log_3(81) = log_3(3^4) = 4. Confirmed."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "logarithm",
                "exponential form"
              ]
            },
            {
              "id": "precalc_ch2_s17",
              "type": "lesson",
              "title": "Exponentials and Logs as Inverses",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Exponential functions and logarithmic functions are inverses of each other. This means they undo each other. If f(x) = b^x, then f^(-1)(x) = log_b(x). Graphically, the log curve is the exponential curve reflected over the line y = x."
                },
                {
                  "kind": "text",
                  "text": "This inverse relationship gives us two cancellation identities: b^(log_b(x)) = x for all x > 0, and log_b(b^x) = x for all real x. These are incredibly useful for solving equations."
                },
                {
                  "kind": "text",
                  "text": "To solve an exponential equation like 5^x = 200, take log base 5 of both sides: x = log_5(200). Using change of base: x = ln(200)/ln(5) \u2248 5.292/1.609 \u2248 3.29. To solve a log equation like ln(x) = 7, exponentiate both sides with base e: x = e^7 \u2248 1096.6."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "exp_log_inverse_reflection"
                }
              ],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_logarithmic_functions",
                "topic_inverse_functions"
              ],
              "glossary_terms": [
                "inverse function",
                "exponential function",
                "logarithm"
              ]
            },
            {
              "id": "precalc_ch2_s18",
              "type": "lesson",
              "title": "Exponentials and Logs as Inverses",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "When you see an equation with the variable in the exponent, think 'take a log.' When the variable is inside a log, think 'exponentiate.' These inverse operations are your main tools."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_logarithmic_functions",
                "topic_inverse_functions"
              ],
              "glossary_terms": [
                "inverse function",
                "exponential function",
                "logarithm"
              ]
            },
            {
              "id": "precalc_ch2_s19",
              "type": "lesson",
              "title": "Solving Exponential and Logarithmic Equations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Many real-world problems require solving equations involving exponentials or logs. The strategy depends on where the variable appears. If x is in the exponent, isolate the exponential expression and take a logarithm of both sides. If x is inside a logarithm, isolate the log and convert to exponential form."
                },
                {
                  "kind": "example",
                  "text": "Solve 4 \u00b7 e^(0.3t) = 100. First isolate: e^(0.3t) = 25. Take ln of both sides: 0.3t = ln(25) \u2248 3.219. Divide: t \u2248 10.73."
                },
                {
                  "kind": "example",
                  "text": "Solve 2\u00b7log(x) + log(3) = log(75). Use log properties: log(x^2) + log(3) = log(75), so log(3x^2) = log(75). Therefore 3x^2 = 75, x^2 = 25, x = 5 (reject -5 since log requires positive input)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "exponential function",
                "logarithm",
                "natural logarithm"
              ]
            },
            {
              "id": "precalc_ch2_s20",
              "type": "lesson",
              "title": "Solving Exponential and Logarithmic Equations",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "Always check your answers in the original equation. Logarithms are only defined for positive inputs, so reject any solution that makes the argument of a log zero or negative."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "exponential function",
                "logarithm",
                "natural logarithm"
              ]
            },
            {
              "id": "precalc_ch2_s21",
              "type": "practice",
              "title": "Practice: Exponentials and Logarithms",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) A bank account has $5,000 growing at 4% per year compounded annually. Write the function A(t) and find the balance after 10 years."
                },
                {
                  "kind": "problem",
                  "text": "2) Evaluate without a calculator: log_4(64), ln(e^5), log(0.001)."
                },
                {
                  "kind": "problem",
                  "text": "3) Use log properties to expand: ln(x^2 \u00b7 sqrt(y) / z^3)."
                },
                {
                  "kind": "problem",
                  "text": "4) Solve for t: 500 \u00b7 (0.9)^t = 200."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_logarithmic_functions",
                "topic_log_properties"
              ],
              "glossary_terms": [
                "exponential function",
                "logarithm",
                "compound interest"
              ]
            },
            {
              "id": "precalc_ch2_s22",
              "type": "summary",
              "title": "Chapter Summary: Exponentials and Logarithms",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Exponential functions f(x) = a \u00b7 b^x model quantities that grow or decay by a fixed percentage per step. The base b determines the behavior: b > 1 gives growth, 0 < b < 1 gives decay. The number e \u2248 2.718 is the natural base for continuous processes."
                },
                {
                  "kind": "text",
                  "text": "Logarithms are the inverses of exponentials: log_b(x) answers 'what power of b gives x?' The three key log properties (product, quotient, power) turn complex expressions into simpler ones. Common logs (base 10) and natural logs (base e) are the most widely used."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "exp_log_chapter_summary"
                }
              ],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "exponential function",
                "logarithm",
                "natural logarithm",
                "Euler's number",
                "inverse function"
              ]
            },
            {
              "id": "precalc_ch2_s23",
              "type": "summary",
              "title": "Chapter Summary: Exponentials and Logarithms",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "To solve exponential equations, take a log. To solve log equations, exponentiate. These inverse operations are the central strategy. Always check that solutions keep log arguments positive. These functions and skills will be essential when you study derivatives and integrals of exponentials and logs in calculus."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_exponential_functions",
                "topic_logarithmic_functions"
              ],
              "glossary_terms": [
                "exponential function",
                "logarithm",
                "natural logarithm",
                "Euler's number",
                "inverse function"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch2_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Exponentials and Logarithms",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch2_test",
            "optional_recommended": true,
            "title": "Chapter Test: Exponentials and Logarithms",
            "questions": []
          }
        },
        {
          "id": "precalc_ch3_trigonometry",
          "title": "Trigonometry",
          "description": "Master angles, the unit circle, sine, cosine, tangent, trig identities, and graphing trig functions \u2014 skills you will use constantly in calculus.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch3_s01",
              "type": "lesson",
              "title": "Angles and Radian Measure",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "An angle measures the amount of rotation from one direction to another. You are probably used to measuring angles in degrees, where a full rotation is 360 degrees. In calculus and advanced math, we use a different unit called radians."
                },
                {
                  "kind": "text",
                  "text": "One radian is the angle created when the arc length along a circle equals the radius of that circle. Since the circumference of a circle is 2\u00b7pi\u00b7r, a full rotation is 2\u00b7pi radians. So 360 degrees = 2\u00b7pi radians, which means 180 degrees = pi radians."
                },
                {
                  "kind": "text",
                  "text": "To convert: multiply degrees by pi/180 to get radians, or multiply radians by 180/pi to get degrees. Key angles to memorize: 30\u00b0 = pi/6, 45\u00b0 = pi/4, 60\u00b0 = pi/3, 90\u00b0 = pi/2, 180\u00b0 = pi, 270\u00b0 = 3\u00b7pi/2, 360\u00b0 = 2\u00b7pi."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "degree_radian_conversion_circle"
                }
              ],
              "related_topic_ids": [
                "topic_angles",
                "topic_radian_measure"
              ],
              "glossary_terms": [
                "angle",
                "radian",
                "degree"
              ]
            },
            {
              "id": "precalc_ch3_s02",
              "type": "lesson",
              "title": "Angles and Radian Measure",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "Calculus almost always uses radians. The derivative formulas for sine and cosine only work cleanly in radians. Get comfortable thinking in radians now \u2014 it will save you headaches later."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_angles",
                "topic_radian_measure"
              ],
              "glossary_terms": [
                "angle",
                "radian",
                "degree"
              ]
            },
            {
              "id": "precalc_ch3_s03",
              "type": "lesson",
              "title": "The Unit Circle",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The unit circle is a circle with radius 1 centered at the origin. It is the key tool for defining trigonometric functions for any angle, not just angles in triangles."
                },
                {
                  "kind": "text",
                  "text": "Start at the point (1, 0) on the right side of the circle. As you move counterclockwise by an angle theta, you arrive at a point on the circle. The x-coordinate of that point is cos(theta) and the y-coordinate is sin(theta). This simple idea defines sine and cosine for all angles."
                },
                {
                  "kind": "text",
                  "text": "Since the circle has radius 1, every point satisfies x^2 + y^2 = 1, which gives us the fundamental identity: cos^2(theta) + sin^2(theta) = 1. This identity is true for every angle and will appear throughout calculus."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "unit_circle_labeled"
                }
              ],
              "related_topic_ids": [
                "topic_unit_circle",
                "topic_trig_functions"
              ],
              "glossary_terms": [
                "unit circle",
                "sine",
                "cosine",
                "Pythagorean identity"
              ]
            },
            {
              "id": "precalc_ch3_s04",
              "type": "lesson",
              "title": "The Unit Circle",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "At theta = pi/3 (60\u00b0), the point on the unit circle is (1/2, sqrt(3)/2). So cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2. Check: (1/2)^2 + (sqrt(3)/2)^2 = 1/4 + 3/4 = 1."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_unit_circle",
                "topic_trig_functions"
              ],
              "glossary_terms": [
                "unit circle",
                "sine",
                "cosine",
                "Pythagorean identity"
              ]
            },
            {
              "id": "precalc_ch3_s05",
              "type": "lesson",
              "title": "Sine, Cosine, and Tangent",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The three primary trig functions are sine, cosine, and tangent. From a right triangle perspective: for an acute angle theta in a right triangle, sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, and tan(theta) = opposite/adjacent. The mnemonic SOH-CAH-TOA helps you remember this."
                },
                {
                  "kind": "text",
                  "text": "Using the unit circle definition: sin(theta) is the y-coordinate, cos(theta) is the x-coordinate, and tan(theta) = sin(theta)/cos(theta) = y/x. Tangent is undefined whenever cos(theta) = 0 (at theta = pi/2, 3\u00b7pi/2, etc.)."
                },
                {
                  "kind": "text",
                  "text": "There are also three reciprocal functions: csc(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), and cot(theta) = 1/tan(theta) = cos(theta)/sin(theta). These show up in calculus integration problems regularly."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "right_triangle_sohcahtoa"
                }
              ],
              "related_topic_ids": [
                "topic_trig_functions"
              ],
              "glossary_terms": [
                "sine",
                "cosine",
                "tangent",
                "cosecant",
                "secant",
                "cotangent",
                "SOH-CAH-TOA"
              ]
            },
            {
              "id": "precalc_ch3_s06",
              "type": "lesson",
              "title": "Sine, Cosine, and Tangent",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "At theta = pi/4 (45\u00b0), the unit circle point is (sqrt(2)/2, sqrt(2)/2). So sin(pi/4) = sqrt(2)/2, cos(pi/4) = sqrt(2)/2, and tan(pi/4) = 1."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_functions"
              ],
              "glossary_terms": [
                "sine",
                "cosine",
                "tangent",
                "cosecant",
                "secant",
                "cotangent",
                "SOH-CAH-TOA"
              ]
            },
            {
              "id": "precalc_ch3_s07",
              "type": "worked_example",
              "title": "Worked Example: Evaluating Trig Functions",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Find the exact values of sin(5\u00b7pi/6), cos(5\u00b7pi/6), and tan(5\u00b7pi/6)."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Find the reference angle. 5\u00b7pi/6 is in the second quadrant (between pi/2 and pi). The reference angle is pi - 5\u00b7pi/6 = pi/6."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Recall the values at pi/6. sin(pi/6) = 1/2 and cos(pi/6) = sqrt(3)/2."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Apply quadrant signs. In the second quadrant, sine is positive and cosine is negative. So sin(5\u00b7pi/6) = 1/2 and cos(5\u00b7pi/6) = -sqrt(3)/2."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "reference_angle_quadrant2"
                }
              ],
              "related_topic_ids": [
                "topic_trig_functions",
                "topic_unit_circle"
              ],
              "glossary_terms": [
                "reference angle",
                "quadrant",
                "sine",
                "cosine",
                "tangent"
              ]
            },
            {
              "id": "precalc_ch3_s08",
              "type": "worked_example",
              "title": "Worked Example: Evaluating Trig Functions",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Compute tangent. tan(5\u00b7pi/6) = sin/cos = (1/2)/(-sqrt(3)/2) = -1/sqrt(3) = -sqrt(3)/3."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_functions",
                "topic_unit_circle"
              ],
              "glossary_terms": [
                "reference angle",
                "quadrant",
                "sine",
                "cosine",
                "tangent"
              ]
            },
            {
              "id": "precalc_ch3_s09",
              "type": "lesson",
              "title": "Graphs of Sine and Cosine",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The graphs of y = sin(x) and y = cos(x) are smooth, wave-like curves that repeat every 2\u00b7pi units. This repeating behavior is called periodicity. The period is the length of one complete cycle: for both basic sine and cosine, the period is 2\u00b7pi."
                },
                {
                  "kind": "text",
                  "text": "Both graphs oscillate between -1 and 1. The amplitude is the distance from the center line (y = 0) to the peak, which is 1 for the basic functions. Sine starts at 0, rises to 1 at pi/2, returns to 0 at pi, drops to -1 at 3\u00b7pi/2, and returns to 0 at 2\u00b7pi. Cosine starts at 1, drops to 0 at pi/2, reaches -1 at pi, returns to 0 at 3\u00b7pi/2, and goes back to 1 at 2\u00b7pi."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "sine_cosine_wave_comparison"
                }
              ],
              "related_topic_ids": [
                "topic_trig_graphs"
              ],
              "glossary_terms": [
                "period",
                "amplitude",
                "sine",
                "cosine",
                "phase shift"
              ]
            },
            {
              "id": "precalc_ch3_s10",
              "type": "lesson",
              "title": "Graphs of Sine and Cosine",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "For the general form y = A\u00b7sin(Bx - C) + D: |A| is the amplitude, 2\u00b7pi/|B| is the period, C/B is the phase shift (horizontal shift), and D is the vertical shift (midline). These parameters let you model any sinusoidal wave."
                },
                {
                  "kind": "example",
                  "text": "y = 3\u00b7sin(2x) has amplitude 3 and period 2\u00b7pi/2 = pi. It oscillates between -3 and 3, completing one full cycle every pi units instead of every 2\u00b7pi units."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_graphs"
              ],
              "glossary_terms": [
                "period",
                "amplitude",
                "sine",
                "cosine",
                "phase shift"
              ]
            },
            {
              "id": "precalc_ch3_s11",
              "type": "lesson",
              "title": "Graphs of Tangent and Other Trig Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The tangent function y = tan(x) has a very different shape from sine and cosine. It has vertical asymptotes at x = pi/2 + n\u00b7pi (where cosine is zero). Between each pair of asymptotes, the graph rises from negative infinity to positive infinity. Its period is pi, not 2\u00b7pi."
                },
                {
                  "kind": "text",
                  "text": "Cotangent y = cot(x) = cos(x)/sin(x) is similar but shifted: it has vertical asymptotes where sine is zero (x = n\u00b7pi) and its period is also pi. The graph falls from positive infinity to negative infinity between asymptotes."
                },
                {
                  "kind": "text",
                  "text": "Secant y = sec(x) = 1/cos(x) has U-shaped branches opening upward and downward, with vertical asymptotes where cosine is zero. Cosecant y = csc(x) = 1/sin(x) has a similar shape but with asymptotes where sine is zero. Both have period 2\u00b7pi."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "tangent_function_graph"
                }
              ],
              "related_topic_ids": [
                "topic_trig_graphs",
                "topic_trig_functions"
              ],
              "glossary_terms": [
                "tangent",
                "cotangent",
                "secant",
                "cosecant",
                "vertical asymptote",
                "period"
              ]
            },
            {
              "id": "precalc_ch3_s12",
              "type": "lesson",
              "title": "Graphs of Tangent and Other Trig Functions",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "You will encounter all six trig functions in calculus. Tangent and secant appear especially often in derivative and integration formulas, so getting comfortable with their graphs and behaviors now is worthwhile."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_graphs",
                "topic_trig_functions"
              ],
              "glossary_terms": [
                "tangent",
                "cotangent",
                "secant",
                "cosecant",
                "vertical asymptote",
                "period"
              ]
            },
            {
              "id": "precalc_ch3_s13",
              "type": "lesson",
              "title": "Fundamental Trigonometric Identities",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Trigonometric identities are equations that are true for all valid angles. They are not something to solve \u2014 they are facts you can always use to simplify or rewrite expressions. Mastering identities is one of the most important parts of pre-calculus."
                },
                {
                  "kind": "text",
                  "text": "Pythagorean identities: sin^2(theta) + cos^2(theta) = 1. Dividing through by cos^2 gives tan^2(theta) + 1 = sec^2(theta). Dividing through by sin^2 gives 1 + cot^2(theta) = csc^2(theta). These three identities come up constantly."
                },
                {
                  "kind": "text",
                  "text": "Reciprocal identities: csc = 1/sin, sec = 1/cos, cot = 1/tan. Quotient identities: tan = sin/cos, cot = cos/sin. Even-odd identities: cos(-theta) = cos(theta) (cosine is even), sin(-theta) = -sin(theta) (sine is odd), tan(-theta) = -tan(theta) (tangent is odd)."
                }
              ],
              "graphics": [
                {
                  "kind": "table",
                  "name": "fundamental_trig_identities_table"
                }
              ],
              "related_topic_ids": [
                "topic_trig_identities"
              ],
              "glossary_terms": [
                "Pythagorean identity",
                "reciprocal identity",
                "even function",
                "odd function"
              ]
            },
            {
              "id": "precalc_ch3_s14",
              "type": "lesson",
              "title": "Fundamental Trigonometric Identities",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "You do not need to memorize all identities blindly. Focus on sin^2 + cos^2 = 1, the reciprocal relationships, and tan = sin/cos. Many other identities can be derived from these."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_identities"
              ],
              "glossary_terms": [
                "Pythagorean identity",
                "reciprocal identity",
                "even function",
                "odd function"
              ]
            },
            {
              "id": "precalc_ch3_s15",
              "type": "lesson",
              "title": "Sum, Difference, and Double Angle Formulas",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The angle sum and difference formulas let you find the trig values of combined angles. sin(A + B) = sin(A)cos(B) + cos(A)sin(B). sin(A - B) = sin(A)cos(B) - cos(A)sin(B). cos(A + B) = cos(A)cos(B) - sin(A)sin(B). cos(A - B) = cos(A)cos(B) + sin(A)sin(B)."
                },
                {
                  "kind": "text",
                  "text": "Setting A = B in the sum formulas gives the double angle formulas. sin(2A) = 2\u00b7sin(A)\u00b7cos(A). cos(2A) = cos^2(A) - sin^2(A), which can also be written as 2\u00b7cos^2(A) - 1 or 1 - 2\u00b7sin^2(A). tan(2A) = 2\u00b7tan(A)/(1 - tan^2(A))."
                },
                {
                  "kind": "text",
                  "text": "From the cos(2A) formulas, we get the half-angle or power-reduction formulas: cos^2(A) = (1 + cos(2A))/2 and sin^2(A) = (1 - cos(2A))/2. These are extremely useful in calculus for integrating sin^2(x) and cos^2(x)."
                }
              ],
              "graphics": [
                {
                  "kind": "table",
                  "name": "angle_sum_difference_formulas"
                }
              ],
              "related_topic_ids": [
                "topic_trig_identities",
                "topic_angle_formulas"
              ],
              "glossary_terms": [
                "angle sum formula",
                "double angle formula",
                "power-reduction formula"
              ]
            },
            {
              "id": "precalc_ch3_s16",
              "type": "lesson",
              "title": "Sum, Difference, and Double Angle Formulas",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Find cos(75\u00b0). Write 75\u00b0 = 45\u00b0 + 30\u00b0. cos(75\u00b0) = cos(45\u00b0)cos(30\u00b0) - sin(45\u00b0)sin(30\u00b0) = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2) = (sqrt(6) - sqrt(2))/4."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_identities",
                "topic_angle_formulas"
              ],
              "glossary_terms": [
                "angle sum formula",
                "double angle formula",
                "power-reduction formula"
              ]
            },
            {
              "id": "precalc_ch3_s17",
              "type": "worked_example",
              "title": "Worked Example: Verifying a Trig Identity",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Verify the identity: (1 - cos^2(x)) / sin(x) = sin(x)."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Focus on the left side. The numerator is 1 - cos^2(x). By the Pythagorean identity, sin^2(x) + cos^2(x) = 1, so 1 - cos^2(x) = sin^2(x)."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Substitute. The left side becomes sin^2(x) / sin(x)."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Simplify. sin^2(x) / sin(x) = sin(x). The left side equals the right side. Identity verified."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_identities"
              ],
              "glossary_terms": [
                "Pythagorean identity",
                "identity"
              ]
            },
            {
              "id": "precalc_ch3_s18",
              "type": "worked_example",
              "title": "Worked Example: Verifying a Trig Identity",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "When verifying identities, work on one side only and transform it into the other side. The Pythagorean identity is your most powerful tool \u2014 look for opportunities to substitute sin^2 + cos^2 = 1 in various rearranged forms."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_identities"
              ],
              "glossary_terms": [
                "Pythagorean identity",
                "identity"
              ]
            },
            {
              "id": "precalc_ch3_s19",
              "type": "lesson",
              "title": "Inverse Trigonometric Functions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Inverse trig functions answer the question: what angle gives this trig value? For example, arcsin(1/2) (also written sin^(-1)(1/2)) asks: what angle has a sine of 1/2? The answer is pi/6 (30\u00b0)."
                },
                {
                  "kind": "text",
                  "text": "Since trig functions are periodic (they repeat), they are not one-to-one on their full domain. To define inverses, we restrict the domain. For arcsin (sin^(-1)), the output is in [-pi/2, pi/2]. For arccos (cos^(-1)), the output is in [0, pi]. For arctan (tan^(-1)), the output is in (-pi/2, pi/2)."
                },
                {
                  "kind": "text",
                  "text": "These restricted ranges ensure each input gives exactly one output. Arcsin has domain [-1, 1], arccos has domain [-1, 1], and arctan has domain all real numbers (with horizontal asymptotes at y = pi/2 and y = -pi/2)."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "inverse_trig_graphs"
                }
              ],
              "related_topic_ids": [
                "topic_inverse_trig"
              ],
              "glossary_terms": [
                "inverse trigonometric function",
                "arcsin",
                "arccos",
                "arctan",
                "restricted domain"
              ]
            },
            {
              "id": "precalc_ch3_s20",
              "type": "lesson",
              "title": "Inverse Trigonometric Functions",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "arccos(-1) = pi because cos(pi) = -1 and pi is in [0, pi]. arctan(1) = pi/4 because tan(pi/4) = 1 and pi/4 is in (-pi/2, pi/2)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_inverse_trig"
              ],
              "glossary_terms": [
                "inverse trigonometric function",
                "arcsin",
                "arccos",
                "arctan",
                "restricted domain"
              ]
            },
            {
              "id": "precalc_ch3_s21",
              "type": "lesson",
              "title": "Solving Trigonometric Equations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A trig equation asks you to find the angle(s) that make a statement true. For example, solve sin(x) = 1/2. From the unit circle, x = pi/6 is one solution. But since sine is periodic, there are infinitely many solutions: x = pi/6 + 2\u00b7n\u00b7pi and x = 5\u00b7pi/6 + 2\u00b7n\u00b7pi for any integer n."
                },
                {
                  "kind": "text",
                  "text": "Most problems ask for solutions in a specific interval like [0, 2\u00b7pi). In that case, list only the solutions that fall in that range. For sin(x) = 1/2 on [0, 2\u00b7pi), the answers are x = pi/6 and x = 5\u00b7pi/6."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "trig_equation_unit_circle_solutions"
                }
              ],
              "related_topic_ids": [
                "topic_trig_equations"
              ],
              "glossary_terms": [
                "trigonometric equation",
                "period",
                "unit circle"
              ]
            },
            {
              "id": "precalc_ch3_s22",
              "type": "lesson",
              "title": "Solving Trigonometric Equations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "More complex trig equations may require algebraic manipulation first. Factor, use identities to rewrite in terms of a single function, or make a substitution. For example, 2\u00b7sin^2(x) - sin(x) - 1 = 0 can be factored as (2\u00b7sin(x) + 1)(sin(x) - 1) = 0, giving sin(x) = -1/2 or sin(x) = 1."
                },
                {
                  "kind": "example",
                  "text": "Solve cos(2x) = 0 on [0, 2\u00b7pi). First, 2x = pi/2 + n\u00b7pi. So x = pi/4 + n\u00b7pi/2. In [0, 2\u00b7pi), that gives x = pi/4, 3\u00b7pi/4, 5\u00b7pi/4, 7\u00b7pi/4."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_equations"
              ],
              "glossary_terms": [
                "trigonometric equation",
                "period",
                "unit circle"
              ]
            },
            {
              "id": "precalc_ch3_s23",
              "type": "practice",
              "title": "Practice: Trigonometry",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) Convert 225\u00b0 to radians and find the exact values of sin(225\u00b0), cos(225\u00b0), and tan(225\u00b0)."
                },
                {
                  "kind": "problem",
                  "text": "2) Verify the identity: tan(x)\u00b7cos(x) = sin(x)."
                },
                {
                  "kind": "problem",
                  "text": "3) Find the amplitude, period, and phase shift of y = 2\u00b7cos(3x - pi)."
                },
                {
                  "kind": "problem",
                  "text": "4) Solve 2\u00b7cos(x) + 1 = 0 on [0, 2\u00b7pi)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_functions",
                "topic_trig_identities",
                "topic_trig_graphs"
              ],
              "glossary_terms": [
                "radian",
                "identity",
                "amplitude",
                "period"
              ]
            },
            {
              "id": "precalc_ch3_s24",
              "type": "summary",
              "title": "Chapter Summary: Trigonometry",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Radians are the standard angle measure in calculus, with pi radians = 180 degrees. The unit circle defines sin(theta) as the y-coordinate and cos(theta) as the x-coordinate of a point on a circle of radius 1."
                },
                {
                  "kind": "text",
                  "text": "The Pythagorean identity sin^2 + cos^2 = 1 is the most fundamental relationship. The angle sum, double angle, and power-reduction formulas expand your ability to simplify and compute. Verifying identities strengthens your algebraic flexibility."
                },
                {
                  "kind": "text",
                  "text": "Trig graphs are periodic waves characterized by amplitude, period, and phase shift. Inverse trig functions (arcsin, arccos, arctan) reverse the process to find angles from values. Solving trig equations requires knowledge of the unit circle and careful attention to periodicity."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "trig_chapter_summary_map"
                }
              ],
              "related_topic_ids": [
                "topic_trig_functions",
                "topic_unit_circle",
                "topic_trig_identities",
                "topic_trig_graphs",
                "topic_inverse_trig"
              ],
              "glossary_terms": [
                "radian",
                "unit circle",
                "sine",
                "cosine",
                "tangent",
                "identity",
                "amplitude",
                "period",
                "inverse trigonometric function"
              ]
            },
            {
              "id": "precalc_ch3_s25",
              "type": "summary",
              "title": "Chapter Summary: Trigonometry",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Trigonometry is woven throughout calculus. Derivatives of sin and cos, trig substitution for integrals, and Fourier analysis all depend on the concepts in this chapter."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_trig_functions",
                "topic_unit_circle",
                "topic_trig_identities",
                "topic_trig_graphs",
                "topic_inverse_trig"
              ],
              "glossary_terms": [
                "radian",
                "unit circle",
                "sine",
                "cosine",
                "tangent",
                "identity",
                "amplitude",
                "period",
                "inverse trigonometric function"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch3_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Trigonometry",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch3_test",
            "optional_recommended": true,
            "title": "Chapter Test: Trigonometry",
            "questions": []
          }
        },
        {
          "id": "precalc_ch4_complex_numbers",
          "title": "Complex Numbers",
          "description": "Learn about the imaginary unit i, how to perform arithmetic with complex numbers, and how they connect to the complex plane and Euler's formula.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch4_s01",
              "type": "lesson",
              "title": "The Imaginary Unit i",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "When you try to solve x^2 = -1, there is no real number that works because squaring any real number gives a positive result (or zero). To handle this, mathematicians invented the imaginary unit i, defined by i^2 = -1. In other words, i = sqrt(-1)."
                },
                {
                  "kind": "text",
                  "text": "With i, you can take square roots of negative numbers. sqrt(-9) = sqrt(9) \u00b7 sqrt(-1) = 3i. sqrt(-25) = 5i. This opens up a whole new number system that extends the real numbers."
                },
                {
                  "kind": "text",
                  "text": "Powers of i follow a repeating cycle of four: i^1 = i, i^2 = -1, i^3 = i^2 \u00b7 i = -i, i^4 = (i^2)^2 = 1. Then it repeats: i^5 = i, i^6 = -1, and so on. To find any power of i, just divide the exponent by 4 and look at the remainder."
                },
                {
                  "kind": "example",
                  "text": "Find i^23. Divide 23 by 4: 23 = 4 \u00b7 5 + 3. The remainder is 3, so i^23 = i^3 = -i."
                }
              ],
              "graphics": [
                {
                  "kind": "table",
                  "name": "powers_of_i_cycle"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers"
              ],
              "glossary_terms": [
                "imaginary unit",
                "complex number"
              ]
            },
            {
              "id": "precalc_ch4_s02",
              "type": "lesson",
              "title": "Complex Number Arithmetic",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A complex number has the form a + bi, where a is the real part and b is the imaginary part. For example, 3 + 2i has real part 3 and imaginary part 2. The number 5 is also complex (5 + 0i), and so is 4i (0 + 4i)."
                },
                {
                  "kind": "text",
                  "text": "Addition and subtraction: combine like parts. (3 + 2i) + (1 - 5i) = 4 - 3i. (7 + i) - (2 + 4i) = 5 - 3i. Just add the real parts together and the imaginary parts together."
                },
                {
                  "kind": "text",
                  "text": "Multiplication: use the distributive property (FOIL). (2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 8 + 10i - 3(-1) = 8 + 10i + 3 = 11 + 10i. The key step is replacing i^2 with -1."
                },
                {
                  "kind": "example",
                  "text": "(1 + i)^2 = 1 + 2i + i^2 = 1 + 2i + (-1) = 2i. Squaring (1 + i) gives a purely imaginary number."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_arithmetic"
              ],
              "glossary_terms": [
                "complex number",
                "real part",
                "imaginary part"
              ]
            },
            {
              "id": "precalc_ch4_s03",
              "type": "lesson",
              "title": "The Complex Plane",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Complex numbers can be visualized on the complex plane (also called the Argand diagram). The horizontal axis represents the real part and the vertical axis represents the imaginary part. The number 3 + 2i is plotted at the point (3, 2)."
                },
                {
                  "kind": "text",
                  "text": "The modulus (or absolute value) of a complex number a + bi is its distance from the origin: |a + bi| = sqrt(a^2 + b^2). This is just the distance formula from the Pythagorean theorem. For example, |3 + 4i| = sqrt(9 + 16) = 5."
                },
                {
                  "kind": "text",
                  "text": "The argument of a complex number is the angle it makes with the positive real axis, measured counterclockwise. For 3 + 4i, the argument is arctan(4/3). Together, modulus and argument give the polar form of a complex number."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "complex_plane_point_plotted"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_plane"
              ],
              "glossary_terms": [
                "complex plane",
                "modulus",
                "argument",
                "Argand diagram"
              ]
            },
            {
              "id": "precalc_ch4_s04",
              "type": "lesson",
              "title": "The Complex Plane",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "The complex plane gives geometry to numbers. Adding complex numbers corresponds to vector addition. Multiplying by i rotates a point 90 degrees counterclockwise. This geometric view becomes powerful in advanced math and engineering."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_plane"
              ],
              "glossary_terms": [
                "complex plane",
                "modulus",
                "argument",
                "Argand diagram"
              ]
            },
            {
              "id": "precalc_ch4_s05",
              "type": "worked_example",
              "title": "Worked Example: Complex Arithmetic",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Let z = 2 - 3i and w = 1 + 4i. Find z + w, z \u00b7 w, and |z|."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Addition. z + w = (2 - 3i) + (1 + 4i) = (2 + 1) + (-3 + 4)i = 3 + i."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Multiplication. z \u00b7 w = (2 - 3i)(1 + 4i) = 2 + 8i - 3i - 12i^2 = 2 + 5i - 12(-1) = 2 + 5i + 12 = 14 + 5i."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Modulus of z. |z| = |2 - 3i| = sqrt(2^2 + (-3)^2) = sqrt(4 + 9) = sqrt(13) \u2248 3.606."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "complex_addition_on_plane"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_arithmetic"
              ],
              "glossary_terms": [
                "complex number",
                "modulus"
              ]
            },
            {
              "id": "precalc_ch4_s06",
              "type": "lesson",
              "title": "Complex Conjugates and Division",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The complex conjugate of a + bi is a - bi. You flip the sign of the imaginary part. For example, the conjugate of 3 + 2i is 3 - 2i. The conjugate of -1 - 5i is -1 + 5i."
                },
                {
                  "kind": "text",
                  "text": "Conjugates have a useful property: when you multiply a complex number by its conjugate, the result is always a real number. (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2. This is because i^2 = -1 makes the cross terms cancel."
                },
                {
                  "kind": "text",
                  "text": "This property is the key to dividing complex numbers. To compute (3 + 2i)/(1 - i), multiply numerator and denominator by the conjugate of the denominator: (3 + 2i)(1 + i) / ((1 - i)(1 + i)) = (3 + 3i + 2i + 2i^2) / (1 + 1) = (3 + 5i - 2) / 2 = (1 + 5i) / 2 = 1/2 + 5i/2."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_conjugate"
              ],
              "glossary_terms": [
                "complex conjugate",
                "rationalize"
              ]
            },
            {
              "id": "precalc_ch4_s07",
              "type": "lesson",
              "title": "Complex Conjugates and Division",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "Multiplying by the conjugate to clear complex denominators is the same idea as rationalizing the denominator with radicals. It is a technique you will use often."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_conjugate"
              ],
              "glossary_terms": [
                "complex conjugate",
                "rationalize"
              ]
            },
            {
              "id": "precalc_ch4_s08",
              "type": "lesson",
              "title": "Polar Form of Complex Numbers",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Instead of writing a complex number as a + bi (rectangular form), you can describe it using its modulus r and argument theta: z = r(cos(theta) + i\u00b7sin(theta)). This is called polar form. Here r = |z| = sqrt(a^2 + b^2) and theta = arctan(b/a) (adjusted for the correct quadrant)."
                },
                {
                  "kind": "text",
                  "text": "Polar form makes multiplication and division beautifully simple. To multiply two complex numbers in polar form, multiply their moduli and add their arguments: r1\u00b7(cos theta1 + i\u00b7sin theta1) times r2\u00b7(cos theta2 + i\u00b7sin theta2) = r1\u00b7r2\u00b7(cos(theta1 + theta2) + i\u00b7sin(theta1 + theta2))."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "polar_form_complex_number"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_polar_form"
              ],
              "glossary_terms": [
                "polar form",
                "modulus",
                "argument",
                "De Moivre's Theorem"
              ]
            },
            {
              "id": "precalc_ch4_s09",
              "type": "lesson",
              "title": "Polar Form of Complex Numbers",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "This also makes computing powers easy. De Moivre's Theorem says: [r(cos theta + i\u00b7sin theta)]^n = r^n \u00b7 (cos(n\u00b7theta) + i\u00b7sin(n\u00b7theta)). To raise a complex number to the nth power, raise the modulus to the nth power and multiply the angle by n."
                },
                {
                  "kind": "example",
                  "text": "Convert 1 + i to polar form. r = sqrt(1 + 1) = sqrt(2). theta = arctan(1/1) = pi/4. So 1 + i = sqrt(2) \u00b7 (cos(pi/4) + i\u00b7sin(pi/4)). Then (1+i)^8 = (sqrt(2))^8 \u00b7 (cos(8\u00b7pi/4) + i\u00b7sin(8\u00b7pi/4)) = 16 \u00b7 (cos(2\u00b7pi) + i\u00b7sin(2\u00b7pi)) = 16."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_polar_form"
              ],
              "glossary_terms": [
                "polar form",
                "modulus",
                "argument",
                "De Moivre's Theorem"
              ]
            },
            {
              "id": "precalc_ch4_s10",
              "type": "worked_example",
              "title": "Worked Example: Converting to Polar Form",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Write z = -1 + sqrt(3)\u00b7i in polar form. Then use De Moivre's Theorem to find z^3."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Find the modulus. r = sqrt((-1)^2 + (sqrt(3))^2) = sqrt(1 + 3) = sqrt(4) = 2."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Find the argument. arctan(sqrt(3)/(-1)) gives a reference angle of pi/3. Since the real part is negative and imaginary part is positive, we are in quadrant II. So theta = pi - pi/3 = 2\u00b7pi/3."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Write polar form. z = 2(cos(2\u00b7pi/3) + i\u00b7sin(2\u00b7pi/3))."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "polar_conversion_quadrant2"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_polar_form"
              ],
              "glossary_terms": [
                "polar form",
                "De Moivre's Theorem",
                "modulus",
                "argument"
              ]
            },
            {
              "id": "precalc_ch4_s11",
              "type": "worked_example",
              "title": "Worked Example: Converting to Polar Form",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Apply De Moivre's Theorem for z^3. z^3 = 2^3 \u00b7 (cos(3 \u00b7 2\u00b7pi/3) + i\u00b7sin(3 \u00b7 2\u00b7pi/3)) = 8 \u00b7 (cos(2\u00b7pi) + i\u00b7sin(2\u00b7pi)) = 8 \u00b7 (1 + 0i) = 8."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_polar_form"
              ],
              "glossary_terms": [
                "polar form",
                "De Moivre's Theorem",
                "modulus",
                "argument"
              ]
            },
            {
              "id": "precalc_ch4_s12",
              "type": "lesson",
              "title": "Euler's Formula Preview",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "One of the most beautiful results in all of mathematics connects exponentials, trigonometry, and complex numbers. Euler's formula states: e^(i\u00b7theta) = cos(theta) + i\u00b7sin(theta). This means the polar form of a complex number can be written as z = r \u00b7 e^(i\u00b7theta)."
                },
                {
                  "kind": "text",
                  "text": "This formula is not obvious \u2014 it will be justified in calculus using Taylor series. But even now, it is worth appreciating. It says that raising e to an imaginary power gives a point on the unit circle in the complex plane. The angle theta determines where on the circle you land."
                },
                {
                  "kind": "text",
                  "text": "A famous special case is Euler's identity: e^(i\u00b7pi) + 1 = 0. This single equation links five of the most important constants in math: e, i, pi, 1, and 0. Many consider it the most elegant equation ever discovered."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "euler_formula_unit_circle"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_euler_formula"
              ],
              "glossary_terms": [
                "Euler's formula",
                "Euler's identity"
              ]
            },
            {
              "id": "precalc_ch4_s13",
              "type": "lesson",
              "title": "Euler's Formula Preview",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "You do not need to fully understand why Euler's formula works yet. Just know it exists and that it creates a deep bridge between exponentials and trig. You will see it again when studying Taylor series in Calculus II."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_euler_formula"
              ],
              "glossary_terms": [
                "Euler's formula",
                "Euler's identity"
              ]
            },
            {
              "id": "precalc_ch4_s14",
              "type": "lesson",
              "title": "Solving Equations with Complex Solutions",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The quadratic formula x = (-b \u00b1 sqrt(b^2 - 4ac))/(2a) sometimes produces a negative number under the square root. When the discriminant b^2 - 4ac is negative, the solutions are complex numbers."
                },
                {
                  "kind": "text",
                  "text": "For example, x^2 + 2x + 5 = 0 gives discriminant = 4 - 20 = -16. So x = (-2 \u00b1 sqrt(-16))/2 = (-2 \u00b1 4i)/2 = -1 \u00b1 2i. The solutions are x = -1 + 2i and x = -1 - 2i."
                },
                {
                  "kind": "text",
                  "text": "Notice the solutions come in conjugate pairs: a + bi and a - bi. This always happens when the polynomial has real coefficients. Complex roots always appear as conjugate pairs, so a real quadratic either has two real roots or two complex conjugate roots."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "parabola_no_real_zeros"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_quadratic_functions"
              ],
              "glossary_terms": [
                "discriminant",
                "complex conjugate",
                "Fundamental Theorem of Algebra"
              ]
            },
            {
              "id": "precalc_ch4_s15",
              "type": "lesson",
              "title": "Solving Equations with Complex Solutions",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "The Fundamental Theorem of Algebra says every polynomial of degree n has exactly n roots when you count complex roots and multiplicities. Complex numbers complete the number system so that every polynomial can be fully factored."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_quadratic_functions"
              ],
              "glossary_terms": [
                "discriminant",
                "complex conjugate",
                "Fundamental Theorem of Algebra"
              ]
            },
            {
              "id": "precalc_ch4_s16",
              "type": "practice",
              "title": "Practice: Complex Numbers",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) Simplify: (4 - 3i)(2 + i)."
                },
                {
                  "kind": "problem",
                  "text": "2) Find the modulus and argument of z = -3 - 3i."
                },
                {
                  "kind": "problem",
                  "text": "3) Divide: (5 + i)/(2 - 3i). Write the result in a + bi form."
                },
                {
                  "kind": "problem",
                  "text": "4) Solve x^2 - 4x + 13 = 0 and express the solutions in a + bi form."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_arithmetic"
              ],
              "glossary_terms": [
                "complex number",
                "modulus",
                "argument",
                "complex conjugate"
              ]
            },
            {
              "id": "precalc_ch4_s17",
              "type": "lesson",
              "title": "Why Complex Numbers Matter",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Complex numbers might seem abstract, but they are indispensable in many fields. In electrical engineering, alternating current is described using complex numbers because voltage and current are sinusoidal waves \u2014 and Euler's formula makes calculations with sinusoids much easier."
                },
                {
                  "kind": "text",
                  "text": "In physics, quantum mechanics is built entirely on complex numbers. The wave function that describes a particle's state is complex-valued. Without complex numbers, there would be no modern understanding of atoms and particles."
                },
                {
                  "kind": "text",
                  "text": "Even in pure math, complex numbers solve problems that seem to have nothing to do with imaginary quantities. They complete the number system, guarantee all polynomials can be factored, and enable powerful techniques in analysis and number theory."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "complex_number_applications"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers"
              ],
              "glossary_terms": [
                "complex number",
                "Euler's formula"
              ]
            },
            {
              "id": "precalc_ch4_s18",
              "type": "lesson",
              "title": "Why Complex Numbers Matter",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "For calculus specifically, complex numbers help explain why certain real-valued integrals work out the way they do, and they connect exponential and trigonometric functions through Euler's formula."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers"
              ],
              "glossary_terms": [
                "complex number",
                "Euler's formula"
              ]
            },
            {
              "id": "precalc_ch4_s19",
              "type": "summary",
              "title": "Chapter Summary: Complex Numbers",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Complex numbers extend the real number system by introducing i where i^2 = -1. Every complex number has the form a + bi with real part a and imaginary part b. Arithmetic follows normal algebra rules with i^2 = -1 applied at the end."
                },
                {
                  "kind": "text",
                  "text": "The complex plane plots these numbers with the real axis horizontal and imaginary axis vertical. Each complex number has a modulus (distance from origin) and argument (angle from positive real axis). Polar form z = r(cos theta + i\u00b7sin theta) simplifies multiplication, division, and powers via De Moivre's Theorem."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "complex_numbers_chapter_summary"
                }
              ],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_plane",
                "topic_euler_formula"
              ],
              "glossary_terms": [
                "complex number",
                "imaginary unit",
                "modulus",
                "argument",
                "complex conjugate",
                "polar form",
                "Euler's formula"
              ]
            },
            {
              "id": "precalc_ch4_s20",
              "type": "summary",
              "title": "Chapter Summary: Complex Numbers",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Complex conjugates (flip the sign of the imaginary part) are used for division and always appear as root pairs of real polynomials. Euler's formula e^(i\u00b7theta) = cos(theta) + i\u00b7sin(theta) unifies exponential and trig functions in a profound way."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_complex_numbers",
                "topic_complex_plane",
                "topic_euler_formula"
              ],
              "glossary_terms": [
                "complex number",
                "imaginary unit",
                "modulus",
                "argument",
                "complex conjugate",
                "polar form",
                "Euler's formula"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch4_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Complex Numbers",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch4_test",
            "optional_recommended": true,
            "title": "Chapter Test: Complex Numbers",
            "questions": []
          }
        },
        {
          "id": "precalc_ch5_sequences_series",
          "title": "Sequences and Series",
          "description": "Learn about ordered patterns of numbers, how to classify and sum them, and build the foundation for the infinite series you will study in Calculus II.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch5_s01",
              "type": "lesson",
              "title": "What Is a Sequence?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the list is called a term. We label them a_1 (first term), a_2 (second term), a_3 (third term), and so on. The subscript tells you the position."
                },
                {
                  "kind": "text",
                  "text": "A sequence can be defined by a formula. For example, a_n = 2n + 1 generates the sequence 3, 5, 7, 9, 11, ... (plug in n = 1, 2, 3, ...). Or a sequence can be defined recursively, where each term depends on previous terms: a_1 = 1, a_n = a_(n-1) + 3 gives 1, 4, 7, 10, ..."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "sequence_convergence_plot"
                }
              ],
              "related_topic_ids": [
                "topic_sequences"
              ],
              "glossary_terms": [
                "sequence",
                "term",
                "converge",
                "diverge"
              ]
            },
            {
              "id": "precalc_ch5_s02",
              "type": "lesson",
              "title": "What Is a Sequence?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "We often ask whether a sequence converges or diverges. A sequence converges if its terms approach a specific number as n gets larger and larger. For example, a_n = 1/n gives 1, 1/2, 1/3, 1/4, ... which approaches 0. A sequence diverges if the terms do not settle on a single value."
                },
                {
                  "kind": "example",
                  "text": "The sequence a_n = (-1)^n gives -1, 1, -1, 1, ... This diverges because it keeps bouncing between -1 and 1 without settling. The sequence a_n = (n+1)/n gives 2, 3/2, 4/3, 5/4, ... which converges to 1."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_sequences"
              ],
              "glossary_terms": [
                "sequence",
                "term",
                "converge",
                "diverge"
              ]
            },
            {
              "id": "precalc_ch5_s03",
              "type": "lesson",
              "title": "Arithmetic Sequences",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference, d. Each term is found by adding d to the previous term: a_n = a_1 + (n-1)\u00b7d."
                },
                {
                  "kind": "text",
                  "text": "For example, 5, 8, 11, 14, 17, ... is arithmetic with a_1 = 5 and d = 3. The nth term is a_n = 5 + (n-1)\u00b73 = 3n + 2. You can use this formula to find any term directly without listing all the ones before it."
                },
                {
                  "kind": "text",
                  "text": "Arithmetic sequences correspond to linear functions. If you plot the terms (n, a_n), they lie on a straight line with slope d. This connection to linearity is why arithmetic sequences feel natural \u2014 they grow at a steady, constant rate."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "arithmetic_sequence_linear_plot"
                }
              ],
              "related_topic_ids": [
                "topic_sequences",
                "topic_arithmetic_sequences"
              ],
              "glossary_terms": [
                "arithmetic sequence",
                "common difference"
              ]
            },
            {
              "id": "precalc_ch5_s04",
              "type": "lesson",
              "title": "Arithmetic Sequences",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Find the 50th term of the sequence 7, 12, 17, 22, ... Here a_1 = 7 and d = 5. a_50 = 7 + (50-1)\u00b75 = 7 + 245 = 252."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_sequences",
                "topic_arithmetic_sequences"
              ],
              "glossary_terms": [
                "arithmetic sequence",
                "common difference"
              ]
            },
            {
              "id": "precalc_ch5_s05",
              "type": "lesson",
              "title": "Geometric Sequences",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A geometric sequence has a constant ratio between consecutive terms. This ratio is called the common ratio, r. Each term is found by multiplying the previous term by r: a_n = a_1 \u00b7 r^(n-1)."
                },
                {
                  "kind": "text",
                  "text": "For example, 3, 6, 12, 24, 48, ... is geometric with a_1 = 3 and r = 2. The nth term is a_n = 3 \u00b7 2^(n-1). Geometric sequences grow (or shrink) multiplicatively rather than additively, so they connect to exponential functions."
                },
                {
                  "kind": "text",
                  "text": "If |r| > 1, the terms grow without bound. If |r| < 1, the terms shrink toward zero. If r is negative, the terms alternate in sign. For instance, r = -1/2 gives a sequence like 8, -4, 2, -1, 1/2, ... which oscillates while shrinking."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "geometric_sequence_exponential_plot"
                }
              ],
              "related_topic_ids": [
                "topic_sequences",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric sequence",
                "common ratio"
              ]
            },
            {
              "id": "precalc_ch5_s06",
              "type": "lesson",
              "title": "Geometric Sequences",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Find the 8th term of 100, 50, 25, ... Here a_1 = 100 and r = 1/2. a_8 = 100 \u00b7 (1/2)^7 = 100 \u00b7 (1/128) = 100/128 = 25/32 \u2248 0.781."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_sequences",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric sequence",
                "common ratio"
              ]
            },
            {
              "id": "precalc_ch5_s07",
              "type": "worked_example",
              "title": "Worked Example: Identifying and Finding Sequence Terms",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "A sequence begins 4, 12, 36, 108, ... Is it arithmetic or geometric? Find the 10th term and a general formula."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Check for arithmetic. Differences: 12 - 4 = 8, 36 - 12 = 24, 108 - 36 = 72. The differences are not constant, so it is not arithmetic."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Check for geometric. Ratios: 12/4 = 3, 36/12 = 3, 108/36 = 3. The ratio is constant at r = 3, so it is geometric."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Write the general formula. a_n = a_1 \u00b7 r^(n-1) = 4 \u00b7 3^(n-1)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_sequences",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric sequence",
                "common ratio",
                "arithmetic sequence"
              ]
            },
            {
              "id": "precalc_ch5_s08",
              "type": "worked_example",
              "title": "Worked Example: Identifying and Finding Sequence Terms",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Find the 10th term. a_10 = 4 \u00b7 3^9 = 4 \u00b7 19683 = 78732."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_sequences",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric sequence",
                "common ratio",
                "arithmetic sequence"
              ]
            },
            {
              "id": "precalc_ch5_s09",
              "type": "lesson",
              "title": "What Is a Series?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A series is the sum of the terms of a sequence. If you have a sequence a_1, a_2, a_3, ..., the corresponding series is S = a_1 + a_2 + a_3 + ... We often want to find the sum of a specific number of terms (a partial sum) or the total of all terms (if the series is infinite)."
                },
                {
                  "kind": "text",
                  "text": "The notation S_n means the sum of the first n terms. For example, if the sequence is 2, 5, 8, 11, then S_4 = 2 + 5 + 8 + 11 = 26."
                },
                {
                  "kind": "text",
                  "text": "Sigma notation (also called summation notation) provides a compact way to write series. The expression \u2211(k=1 to n) a_k means add up a_k for k = 1, 2, 3, ..., n. The letter k is the index of summation, and it takes on each integer value from the lower bound to the upper bound."
                },
                {
                  "kind": "example",
                  "text": "\u2211(k=1 to 4) k^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "sigma_notation_parts"
                }
              ],
              "related_topic_ids": [
                "topic_series",
                "topic_sigma_notation"
              ],
              "glossary_terms": [
                "series",
                "partial sum",
                "sigma notation",
                "index of summation"
              ]
            },
            {
              "id": "precalc_ch5_s10",
              "type": "lesson",
              "title": "Arithmetic Series",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "An arithmetic series is the sum of an arithmetic sequence. There is a simple formula: the sum of the first n terms is S_n = n/2 \u00b7 (a_1 + a_n), or equivalently S_n = n/2 \u00b7 (2a_1 + (n-1)d)."
                },
                {
                  "kind": "text",
                  "text": "Why does this work? The idea comes from pairing terms. In the sum 1 + 2 + 3 + ... + 100, pair the first and last: 1 + 100 = 101. Pair the second and second-to-last: 2 + 99 = 101. There are 50 such pairs, so the total is 50 \u00b7 101 = 5050. This is the method Gauss reportedly discovered as a child."
                },
                {
                  "kind": "text",
                  "text": "The formula works for any arithmetic series. You just need a_1, a_n (or d), and n. The sum grows quadratically with n because you are adding more and more terms that are themselves getting larger."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "arithmetic_series_pairing"
                }
              ],
              "related_topic_ids": [
                "topic_series",
                "topic_arithmetic_sequences"
              ],
              "glossary_terms": [
                "arithmetic series",
                "partial sum"
              ]
            },
            {
              "id": "precalc_ch5_s11",
              "type": "lesson",
              "title": "Arithmetic Series",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Find the sum of the first 20 terms of 5, 9, 13, 17, ... Here a_1 = 5, d = 4, n = 20. a_20 = 5 + 19\u00b74 = 81. S_20 = 20/2 \u00b7 (5 + 81) = 10 \u00b7 86 = 860."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_series",
                "topic_arithmetic_sequences"
              ],
              "glossary_terms": [
                "arithmetic series",
                "partial sum"
              ]
            },
            {
              "id": "precalc_ch5_s12",
              "type": "lesson",
              "title": "Geometric Series",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A geometric series is the sum of a geometric sequence. The sum of the first n terms is: S_n = a_1 \u00b7 (1 - r^n)/(1 - r), valid when r \u2260 1. This formula is derived by a clever trick: multiply S_n by r, then subtract to cancel most terms."
                },
                {
                  "kind": "text",
                  "text": "When |r| < 1, the terms get smaller and smaller. As n approaches infinity, r^n approaches 0, and the infinite geometric series has a finite sum: S = a_1 / (1 - r). This is your first example of an infinite process producing a finite result \u2014 a concept central to calculus."
                },
                {
                  "kind": "text",
                  "text": "When |r| >= 1, the terms do not shrink, and the infinite series diverges (the sum grows without bound or oscillates). So the condition |r| < 1 is critical for convergence."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "geometric_series_partial_sums"
                }
              ],
              "related_topic_ids": [
                "topic_series",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric series",
                "common ratio",
                "converge",
                "diverge",
                "infinite series"
              ]
            },
            {
              "id": "precalc_ch5_s13",
              "type": "lesson",
              "title": "Geometric Series",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Sum the infinite series 10 + 5 + 2.5 + 1.25 + ... Here a_1 = 10 and r = 1/2. Since |r| < 1, S = 10/(1 - 1/2) = 10/(1/2) = 20. The infinite sum is exactly 20."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_series",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric series",
                "common ratio",
                "converge",
                "diverge",
                "infinite series"
              ]
            },
            {
              "id": "precalc_ch5_s14",
              "type": "worked_example",
              "title": "Worked Example: Sum of a Geometric Series",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "A ball is dropped from 6 feet. Each time it bounces, it reaches 2/3 of its previous height. Find the total vertical distance the ball travels (up and down combined) before coming to rest."
                },
                {
                  "kind": "step",
                  "text": "Step 1: The ball falls 6 feet initially. Then it bounces up 6\u00b7(2/3) = 4 feet and falls 4 feet. Then it bounces up 4\u00b7(2/3) = 8/3 feet and falls 8/3 feet. Each bounce adds both an up and a down distance."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Total distance = 6 (initial drop) + 2\u00b7(4 + 8/3 + 16/9 + ...). The factor of 2 accounts for going up and coming back down on each bounce."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "bouncing_ball_geometric"
                }
              ],
              "related_topic_ids": [
                "topic_series",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric series",
                "infinite series",
                "converge"
              ]
            },
            {
              "id": "precalc_ch5_s15",
              "type": "worked_example",
              "title": "Worked Example: Sum of a Geometric Series",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 3: The bouncing heights form an infinite geometric series with a_1 = 4 and r = 2/3. Sum = 4/(1 - 2/3) = 4/(1/3) = 12."
                },
                {
                  "kind": "step",
                  "text": "Step 4: Total distance = 6 + 2\u00b712 = 6 + 24 = 30 feet. The ball travels 30 feet total before stopping."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_series",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "geometric series",
                "infinite series",
                "converge"
              ]
            },
            {
              "id": "precalc_ch5_s16",
              "type": "lesson",
              "title": "Sigma Notation in Depth",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Sigma notation \u2211 is the standard way to write sums compactly. It has three parts: the expression being summed (like k^2 or 3\u00b7(1/2)^k), the starting index (written below \u2211, like k=1), and the ending index (written above \u2211, like n or infinity)."
                },
                {
                  "kind": "text",
                  "text": "You can manipulate sums using properties: \u2211(k=1 to n) (a_k + b_k) = \u2211 a_k + \u2211 b_k. And \u2211(k=1 to n) c\u00b7a_k = c \u00b7 \u2211 a_k. These let you break complex sums into simpler pieces."
                },
                {
                  "kind": "text",
                  "text": "Some useful closed-form sums to know: \u2211(k=1 to n) k = n(n+1)/2. \u2211(k=1 to n) k^2 = n(n+1)(2n+1)/6. \u2211(k=1 to n) k^3 = [n(n+1)/2]^2. These formulas are used when computing Riemann sums in calculus."
                },
                {
                  "kind": "example",
                  "text": "Compute \u2211(k=1 to 5) (2k - 1) = 1 + 3 + 5 + 7 + 9 = 25. Alternatively, using the formula: 2\u00b7\u2211k - \u22111 = 2\u00b7(5\u00b76/2) - 5 = 30 - 5 = 25."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "sigma_notation_anatomy"
                }
              ],
              "related_topic_ids": [
                "topic_sigma_notation"
              ],
              "glossary_terms": [
                "sigma notation",
                "index of summation",
                "closed-form"
              ]
            },
            {
              "id": "precalc_ch5_s17",
              "type": "lesson",
              "title": "Infinite Series and the Idea of Convergence",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "When we write an infinite series like \u2211(k=1 to infinity) a_k, we are asking: does adding up all these terms give a finite number? If so, the series converges to that number. If not, it diverges."
                },
                {
                  "kind": "text",
                  "text": "Formally, we look at the partial sums S_n = a_1 + a_2 + ... + a_n. If the sequence of partial sums S_1, S_2, S_3, ... approaches a finite limit L as n goes to infinity, then the series converges to L. This is the same limit concept you will formalize in Calculus I."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "partial_sums_convergence_divergence"
                }
              ],
              "related_topic_ids": [
                "topic_series",
                "topic_convergence"
              ],
              "glossary_terms": [
                "infinite series",
                "converge",
                "diverge",
                "partial sum",
                "harmonic series"
              ]
            },
            {
              "id": "precalc_ch5_s18",
              "type": "lesson",
              "title": "Infinite Series and the Idea of Convergence",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A necessary condition for convergence is that the terms a_n must approach zero. If a_n does not go to zero, the series definitely diverges. But be careful: a_n going to zero is necessary, not sufficient. The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... has terms going to zero, but the sum still diverges to infinity."
                },
                {
                  "kind": "note",
                  "text": "In Calculus II, you will learn many tests to determine whether series converge or diverge. For now, just understand the core idea: convergence means the partial sums settle to a finite value."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_series",
                "topic_convergence"
              ],
              "glossary_terms": [
                "infinite series",
                "converge",
                "diverge",
                "partial sum",
                "harmonic series"
              ]
            },
            {
              "id": "precalc_ch5_s19",
              "type": "practice",
              "title": "Practice: Sequences and Series",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) Find a formula for the nth term of the sequence 2, 7, 12, 17, 22, ... and find a_30."
                },
                {
                  "kind": "problem",
                  "text": "2) A geometric sequence has a_1 = 81 and r = 1/3. Find the 6th term."
                },
                {
                  "kind": "problem",
                  "text": "3) Find the sum of the first 15 terms of the arithmetic series 3 + 7 + 11 + 15 + ..."
                },
                {
                  "kind": "problem",
                  "text": "4) Determine whether the infinite geometric series 12 + 4 + 4/3 + 4/9 + ... converges. If so, find the sum."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_sequences",
                "topic_series"
              ],
              "glossary_terms": [
                "arithmetic sequence",
                "geometric sequence",
                "series",
                "converge"
              ]
            },
            {
              "id": "precalc_ch5_s20",
              "type": "summary",
              "title": "Chapter Summary: Sequences and Series",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A sequence is an ordered list of numbers. Arithmetic sequences have a constant difference d between terms (a_n = a_1 + (n-1)d). Geometric sequences have a constant ratio r (a_n = a_1 \u00b7 r^(n-1)). Sequences may converge to a limit or diverge."
                },
                {
                  "kind": "text",
                  "text": "A series is the sum of a sequence's terms. The arithmetic series sum is S_n = n/2 \u00b7 (a_1 + a_n). The geometric series sum is S_n = a_1 \u00b7 (1 - r^n)/(1 - r). Sigma notation provides compact shorthand for writing sums."
                },
                {
                  "kind": "text",
                  "text": "An infinite geometric series converges to a_1/(1 - r) when |r| < 1. This is one of the first examples of an infinite process producing a finite result. The concepts of convergence and divergence introduced here will be studied rigorously in Calculus I (limits) and Calculus II (infinite series)."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "sequences_series_chapter_summary"
                }
              ],
              "related_topic_ids": [
                "topic_sequences",
                "topic_series",
                "topic_arithmetic_sequences",
                "topic_geometric_sequences"
              ],
              "glossary_terms": [
                "sequence",
                "series",
                "arithmetic sequence",
                "geometric sequence",
                "converge",
                "diverge",
                "sigma notation"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch5_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Sequences and Series",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch5_test",
            "optional_recommended": true,
            "title": "Chapter Test: Sequences and Series",
            "questions": []
          }
        },
        {
          "id": "precalc_ch6_conics",
          "title": "Conic Sections and Analytic Geometry",
          "description": "Explore the equations and graphs of parabolas, ellipses, and hyperbolas. Strengthen your graphing skills and understand these curves that appear in physics, astronomy, and engineering.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch6_s01",
              "type": "lesson",
              "title": "What Are Conic Sections?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Conic sections are the curves you get when you slice a double cone (two cones tip-to-tip) with a flat plane. Depending on the angle of the cut, you get a circle, an ellipse, a parabola, or a hyperbola. These four shapes have been studied for over 2000 years and appear throughout science and engineering."
                },
                {
                  "kind": "text",
                  "text": "A circle is the simplest conic \u2014 a perfectly round curve where every point is the same distance from the center. An ellipse is a stretched circle, like an oval. A parabola is an open curve with one branch (think of a satellite dish). A hyperbola has two separate branches that open in opposite directions."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "cone_slicing_four_conics"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections"
              ],
              "glossary_terms": [
                "conic section",
                "circle",
                "ellipse",
                "parabola",
                "hyperbola"
              ]
            },
            {
              "id": "precalc_ch6_s02",
              "type": "lesson",
              "title": "What Are Conic Sections?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Each conic section has a standard equation that describes its shape, and special points (like the center, focus, or vertex) that define its geometry. Understanding these equations lets you quickly sketch the curve and identify its key features."
                },
                {
                  "kind": "note",
                  "text": "You already met parabolas as graphs of quadratic functions. Now you will see them as part of a larger family of curves, and you will learn about ellipses and hyperbolas for the first time."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections"
              ],
              "glossary_terms": [
                "conic section",
                "circle",
                "ellipse",
                "parabola",
                "hyperbola"
              ]
            },
            {
              "id": "precalc_ch6_s03",
              "type": "lesson",
              "title": "Circles",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A circle is the set of all points at a fixed distance r (the radius) from a fixed point (h, k) called the center. Its standard equation is (x - h)^2 + (y - k)^2 = r^2."
                },
                {
                  "kind": "text",
                  "text": "When the center is at the origin, this simplifies to x^2 + y^2 = r^2. For example, x^2 + y^2 = 25 is a circle centered at (0, 0) with radius 5."
                },
                {
                  "kind": "text",
                  "text": "If the equation is given in expanded form like x^2 + y^2 - 6x + 4y - 12 = 0, you can convert to standard form by completing the square for both x and y. Group the x terms and y terms, complete the square for each, and you will reveal the center and radius."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "circle_standard_form"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_circles"
              ],
              "glossary_terms": [
                "circle",
                "radius",
                "center",
                "completing the square"
              ]
            },
            {
              "id": "precalc_ch6_s04",
              "type": "lesson",
              "title": "Circles",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Complete the square: x^2 - 6x + y^2 + 4y = 12. For x: (x^2 - 6x + 9) = (x-3)^2, add 9 to both sides. For y: (y^2 + 4y + 4) = (y+2)^2, add 4 to both sides. Result: (x-3)^2 + (y+2)^2 = 25. Center (3, -2), radius 5."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_circles"
              ],
              "glossary_terms": [
                "circle",
                "radius",
                "center",
                "completing the square"
              ]
            },
            {
              "id": "precalc_ch6_s05",
              "type": "lesson",
              "title": "Ellipses",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "An ellipse looks like a stretched circle. It has two special points inside called foci (singular: focus). The defining property of an ellipse is: for any point on the ellipse, the sum of the distances to the two foci is constant."
                },
                {
                  "kind": "text",
                  "text": "The standard equation of an ellipse centered at the origin is x^2/a^2 + y^2/b^2 = 1. Here a and b are the semi-axes. If a > b, the ellipse is wider than it is tall (the major axis is horizontal). If b > a, it is taller than wide (the major axis is vertical)."
                },
                {
                  "kind": "text",
                  "text": "The major axis has length 2a (or 2b, whichever is larger) and connects the two vertices. The minor axis has length 2b (or 2a, whichever is smaller). The foci lie along the major axis at distance c from the center, where c^2 = a^2 - b^2 (assuming a > b)."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "ellipse_labeled_axes_foci"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_ellipses"
              ],
              "glossary_terms": [
                "ellipse",
                "focus",
                "foci",
                "semi-major axis",
                "semi-minor axis",
                "vertex"
              ]
            },
            {
              "id": "precalc_ch6_s06",
              "type": "lesson",
              "title": "Ellipses",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "x^2/25 + y^2/9 = 1 has a = 5, b = 3. Major axis is horizontal (since 5 > 3). c = sqrt(25 - 9) = sqrt(16) = 4. Foci at (4, 0) and (-4, 0). Vertices at (5, 0) and (-5, 0)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_ellipses"
              ],
              "glossary_terms": [
                "ellipse",
                "focus",
                "foci",
                "semi-major axis",
                "semi-minor axis",
                "vertex"
              ]
            },
            {
              "id": "precalc_ch6_s07",
              "type": "lesson",
              "title": "Hyperbolas",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A hyperbola consists of two separate branches that open in opposite directions. Like an ellipse, it has two foci. But the defining property is different: for any point on the hyperbola, the difference of the distances to the two foci is constant."
                },
                {
                  "kind": "text",
                  "text": "The standard equation for a hyperbola centered at the origin opening left and right is x^2/a^2 - y^2/b^2 = 1. For one opening up and down, it is y^2/a^2 - x^2/b^2 = 1. Notice the subtraction sign \u2014 that is what makes it a hyperbola instead of an ellipse."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "hyperbola_with_asymptotes"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_hyperbolas"
              ],
              "glossary_terms": [
                "hyperbola",
                "asymptote",
                "focus",
                "foci",
                "vertex"
              ]
            },
            {
              "id": "precalc_ch6_s08",
              "type": "lesson",
              "title": "Hyperbolas",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Hyperbolas have asymptotes \u2014 lines that the branches approach but never touch. For x^2/a^2 - y^2/b^2 = 1, the asymptotes are the lines y = (b/a)x and y = -(b/a)x. These asymptotes form an X through the center and guide the shape of the curve. The foci are at distance c from the center, where c^2 = a^2 + b^2."
                },
                {
                  "kind": "example",
                  "text": "x^2/16 - y^2/9 = 1 opens left-right. a = 4, b = 3. Asymptotes: y = (3/4)x and y = -(3/4)x. c = sqrt(16 + 9) = 5. Foci at (5, 0) and (-5, 0). Vertices at (4, 0) and (-4, 0)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_hyperbolas"
              ],
              "glossary_terms": [
                "hyperbola",
                "asymptote",
                "focus",
                "foci",
                "vertex"
              ]
            },
            {
              "id": "precalc_ch6_s09",
              "type": "worked_example",
              "title": "Worked Example: Identifying a Conic Section",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Identify the conic section given by the equation 9x^2 + 4y^2 = 36. Find the center, vertices, and foci."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Divide both sides by 36 to get standard form: x^2/4 + y^2/9 = 1."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Both terms are positive with a plus sign and the right side is 1. This is an ellipse. Since 9 > 4, the larger denominator is under y^2, so the major axis is vertical."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Read off the values. Under y^2: b^2 = 9, so b = 3. Under x^2: a^2 = 4, so a = 2. (Here we use the convention that b is the semi-major axis since b > a for this vertical ellipse.)"
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "ellipse_vertical_major_axis"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_ellipses"
              ],
              "glossary_terms": [
                "ellipse",
                "vertex",
                "focus",
                "semi-major axis"
              ]
            },
            {
              "id": "precalc_ch6_s10",
              "type": "worked_example",
              "title": "Worked Example: Identifying a Conic Section",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Center is (0, 0). Vertices on the major axis: (0, 3) and (0, -3). c^2 = 9 - 4 = 5, so c = sqrt(5) \u2248 2.24. Foci at (0, sqrt(5)) and (0, -sqrt(5))."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_ellipses"
              ],
              "glossary_terms": [
                "ellipse",
                "vertex",
                "focus",
                "semi-major axis"
              ]
            },
            {
              "id": "precalc_ch6_s11",
              "type": "lesson",
              "title": "Parabolas Revisited: Focus and Directrix",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "You already know parabolas as graphs of quadratic functions. Now we look at them as conic sections with a geometric definition: a parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix)."
                },
                {
                  "kind": "text",
                  "text": "For a parabola opening upward with vertex at the origin, the equation is x^2 = 4py, where p is the distance from the vertex to the focus. The focus is at (0, p) and the directrix is the horizontal line y = -p."
                },
                {
                  "kind": "text",
                  "text": "If the parabola opens rightward, the equation becomes y^2 = 4px, with focus at (p, 0) and directrix x = -p. Parabolas can open in any of the four directions depending on the equation form."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "parabola_focus_directrix"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_parabolas"
              ],
              "glossary_terms": [
                "parabola",
                "focus",
                "directrix",
                "vertex"
              ]
            },
            {
              "id": "precalc_ch6_s12",
              "type": "lesson",
              "title": "Parabolas Revisited: Focus and Directrix",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "For x^2 = 12y, we have 4p = 12, so p = 3. The vertex is (0,0), the focus is (0, 3), and the directrix is y = -3. The parabola opens upward."
                },
                {
                  "kind": "note",
                  "text": "The focus-directrix property of parabolas is why satellite dishes and car headlights are parabolic \u2014 signals hitting the dish parallel to its axis all reflect to the focus, concentrating energy at one point."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_parabolas"
              ],
              "glossary_terms": [
                "parabola",
                "focus",
                "directrix",
                "vertex"
              ]
            },
            {
              "id": "precalc_ch6_s13",
              "type": "lesson",
              "title": "Eccentricity and Comparing Conics",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "All conic sections can be described by a single number called eccentricity, written e (not to be confused with Euler's number). Eccentricity measures how much the conic deviates from being a circle."
                },
                {
                  "kind": "text",
                  "text": "For a circle, e = 0 (perfectly round). For an ellipse, 0 < e < 1 (the closer to 0, the more circular). For a parabola, e = 1 exactly. For a hyperbola, e > 1. Eccentricity is computed as e = c/a for ellipses and hyperbolas."
                },
                {
                  "kind": "text",
                  "text": "This gives a unified view: as eccentricity increases from 0 past 1, you move from circles to ellipses to parabolas to hyperbolas. This unifying idea appears in orbital mechanics \u2014 planets orbit in ellipses, comets can follow parabolic or hyperbolic paths."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "eccentricity_spectrum_conics"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections"
              ],
              "glossary_terms": [
                "eccentricity",
                "conic section"
              ]
            },
            {
              "id": "precalc_ch6_s14",
              "type": "lesson",
              "title": "Eccentricity and Comparing Conics",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Earth's orbit has eccentricity about 0.017 (very close to a circle). A comet on a one-time visit might have e = 1.2 (hyperbolic \u2014 it flies by the sun and never returns)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections"
              ],
              "glossary_terms": [
                "eccentricity",
                "conic section"
              ]
            },
            {
              "id": "precalc_ch6_s15",
              "type": "lesson",
              "title": "The Distance and Midpoint Formulas",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Analytic geometry uses algebra to study geometric shapes on the coordinate plane. Two of the most fundamental tools are the distance formula and the midpoint formula."
                },
                {
                  "kind": "text",
                  "text": "The distance between two points (x1, y1) and (x2, y2) is d = sqrt((x2 - x1)^2 + (y2 - y1)^2). This comes directly from the Pythagorean theorem: the horizontal and vertical differences form the legs of a right triangle, and the distance is the hypotenuse."
                },
                {
                  "kind": "text",
                  "text": "The midpoint of a line segment connecting (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). You simply average the x-coordinates and average the y-coordinates."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "distance_formula_right_triangle"
                }
              ],
              "related_topic_ids": [
                "topic_analytic_geometry"
              ],
              "glossary_terms": [
                "distance formula",
                "midpoint formula",
                "Pythagorean theorem"
              ]
            },
            {
              "id": "precalc_ch6_s16",
              "type": "lesson",
              "title": "The Distance and Midpoint Formulas",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Find the distance and midpoint between (1, 3) and (7, 11). Distance: sqrt((7-1)^2 + (11-3)^2) = sqrt(36 + 64) = sqrt(100) = 10. Midpoint: ((1+7)/2, (3+11)/2) = (4, 7)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_analytic_geometry"
              ],
              "glossary_terms": [
                "distance formula",
                "midpoint formula",
                "Pythagorean theorem"
              ]
            },
            {
              "id": "precalc_ch6_s17",
              "type": "worked_example",
              "title": "Worked Example: Graphing a Hyperbola",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Graph the hyperbola given by (y - 1)^2/4 - (x + 2)^2/9 = 1. Identify the center, vertices, foci, and asymptotes."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Identify the center. The equation is in standard form with center (h, k) = (-2, 1), read from (x - (-2)) and (y - 1)."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Determine orientation. The positive term is (y - 1)^2, so the hyperbola opens up and down (vertical transverse axis). a^2 = 4 so a = 2, b^2 = 9 so b = 3."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Find vertices. They are a units above and below the center: (-2, 1+2) = (-2, 3) and (-2, 1-2) = (-2, -1)."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "hyperbola_vertical_shifted"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_hyperbolas"
              ],
              "glossary_terms": [
                "hyperbola",
                "vertex",
                "focus",
                "asymptote",
                "transverse axis"
              ]
            },
            {
              "id": "precalc_ch6_s18",
              "type": "worked_example",
              "title": "Worked Example: Graphing a Hyperbola",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Find foci. c^2 = a^2 + b^2 = 4 + 9 = 13, so c = sqrt(13) \u2248 3.61. Foci at (-2, 1 + sqrt(13)) and (-2, 1 - sqrt(13))."
                },
                {
                  "kind": "step",
                  "text": "Step 5: Asymptotes. For a vertical hyperbola centered at (h, k): y - k = \u00b1(a/b)(x - h). So y - 1 = \u00b1(2/3)(x + 2), or y = (2/3)x + 7/3 and y = -(2/3)x - 1/3."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_hyperbolas"
              ],
              "glossary_terms": [
                "hyperbola",
                "vertex",
                "focus",
                "asymptote",
                "transverse axis"
              ]
            },
            {
              "id": "precalc_ch6_s19",
              "type": "lesson",
              "title": "General Second-Degree Equations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Any conic section can be described by the general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. You can determine which conic it is by looking at the coefficients."
                },
                {
                  "kind": "text",
                  "text": "When B = 0 (no xy term): if A = C, it is a circle. If A and C have the same sign but are different, it is an ellipse. If A or C is zero (but not both), it is a parabola. If A and C have opposite signs, it is a hyperbola."
                },
                {
                  "kind": "text",
                  "text": "When there is a B term (Bxy), the conic is rotated. The discriminant B^2 - 4AC determines the type: negative means ellipse or circle, zero means parabola, positive means hyperbola. Rotation of conics is an advanced topic, but knowing the discriminant test is useful."
                }
              ],
              "graphics": [
                {
                  "kind": "table",
                  "name": "conic_classification_table"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections"
              ],
              "glossary_terms": [
                "conic section",
                "discriminant"
              ]
            },
            {
              "id": "precalc_ch6_s20",
              "type": "lesson",
              "title": "General Second-Degree Equations",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Classify 3x^2 - 5y^2 + 6x + 20y - 7 = 0. Here A = 3, B = 0, C = -5. A and C have opposite signs, so this is a hyperbola."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections"
              ],
              "glossary_terms": [
                "conic section",
                "discriminant"
              ]
            },
            {
              "id": "precalc_ch6_s21",
              "type": "practice",
              "title": "Practice: Conic Sections",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) Write the equation of a circle with center (-1, 4) and radius 6."
                },
                {
                  "kind": "problem",
                  "text": "2) For the ellipse x^2/49 + y^2/16 = 1, find the vertices, co-vertices, and foci."
                },
                {
                  "kind": "problem",
                  "text": "3) Find the vertex, focus, and directrix of the parabola y^2 = -8x."
                },
                {
                  "kind": "problem",
                  "text": "4) Classify the conic: 4x^2 + 4y^2 - 16x + 8y + 11 = 0. Rewrite in standard form."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_circles",
                "topic_ellipses",
                "topic_parabolas"
              ],
              "glossary_terms": [
                "circle",
                "ellipse",
                "parabola",
                "vertex",
                "focus",
                "directrix"
              ]
            },
            {
              "id": "precalc_ch6_s22",
              "type": "summary",
              "title": "Chapter Summary: Conic Sections",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Conic sections \u2014 circles, ellipses, parabolas, and hyperbolas \u2014 arise from slicing a cone. Each has a standard equation and key features (center, vertices, foci, asymptotes, directrix) that define its shape."
                },
                {
                  "kind": "text",
                  "text": "Circles: (x-h)^2 + (y-k)^2 = r^2. Ellipses: x^2/a^2 + y^2/b^2 = 1 with c^2 = a^2 - b^2. Parabolas: x^2 = 4py (or y^2 = 4px). Hyperbolas: x^2/a^2 - y^2/b^2 = 1 with c^2 = a^2 + b^2."
                },
                {
                  "kind": "text",
                  "text": "Eccentricity unifies conics: e = 0 for circles, 0 < e < 1 for ellipses, e = 1 for parabolas, e > 1 for hyperbolas. The distance and midpoint formulas are fundamental tools. Conics appear in planetary orbits, satellite dishes, bridge arches, and many other real-world applications."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "conic_sections_summary_chart"
                }
              ],
              "related_topic_ids": [
                "topic_conic_sections",
                "topic_circles",
                "topic_ellipses",
                "topic_parabolas",
                "topic_hyperbolas"
              ],
              "glossary_terms": [
                "conic section",
                "circle",
                "ellipse",
                "parabola",
                "hyperbola",
                "eccentricity",
                "focus",
                "vertex",
                "asymptote"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch6_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Conic Sections",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch6_test",
            "optional_recommended": true,
            "title": "Chapter Test: Conic Sections",
            "questions": []
          }
        },
        {
          "id": "precalc_ch7_vectors_parametric",
          "title": "Vectors and Parametric Equations",
          "description": "Explore vectors in the plane, learn vector operations, and discover how parametric equations describe curves and motion \u2014 essential tools for multivariable calculus.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch7_s01",
              "type": "lesson",
              "title": "What Is a Vector?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A vector is a quantity that has both magnitude (size) and direction. Think of it as an arrow: the length of the arrow represents how much, and the direction the arrow points tells you which way."
                },
                {
                  "kind": "text",
                  "text": "Examples of vectors in real life: velocity (speed and direction of movement), force (strength and direction of a push), and displacement (how far and in what direction you moved). Compare this to a scalar, which has only magnitude \u2014 like temperature, mass, or speed without direction."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "vector_arrow_components"
                }
              ],
              "related_topic_ids": [
                "topic_vectors"
              ],
              "glossary_terms": [
                "vector",
                "scalar",
                "magnitude",
                "component form"
              ]
            },
            {
              "id": "precalc_ch7_s02",
              "type": "lesson",
              "title": "What Is a Vector?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "We write vectors using component form: v = <a, b> means the vector points a units in the x-direction and b units in the y-direction. The vector <3, 4> starts at the origin (by convention) and ends at the point (3, 4). Vectors can be placed anywhere \u2014 only the magnitude and direction matter, not the starting point."
                },
                {
                  "kind": "example",
                  "text": "A car drives 5 miles east and 2 miles north. Its displacement vector is <5, 2>. The magnitude (distance traveled in a straight line) is sqrt(5^2 + 2^2) = sqrt(29) \u2248 5.39 miles."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_vectors"
              ],
              "glossary_terms": [
                "vector",
                "scalar",
                "magnitude",
                "component form"
              ]
            },
            {
              "id": "precalc_ch7_s03",
              "type": "lesson",
              "title": "Vector Operations: Addition and Scalar Multiplication",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Vector addition combines two vectors by adding their components. If u = <a, b> and v = <c, d>, then u + v = <a + c, b + d>. Geometrically, this is the 'tip-to-tail' method: place the tail of v at the tip of u, and the sum is the arrow from the start of u to the tip of v."
                },
                {
                  "kind": "text",
                  "text": "Scalar multiplication scales a vector. If k is a number and v = <a, b>, then k\u00b7v = <ka, kb>. If k > 1, the vector stretches. If 0 < k < 1, it shrinks. If k < 0, it flips direction and scales."
                },
                {
                  "kind": "text",
                  "text": "Vector subtraction is u - v = u + (-v) = <a - c, b - d>. Geometrically, u - v points from the tip of v to the tip of u (when both start at the same point)."
                },
                {
                  "kind": "example",
                  "text": "If u = <2, 5> and v = <-1, 3>: u + v = <1, 8>, u - v = <3, 2>, and 3u = <6, 15>."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "vector_addition_tip_to_tail"
                }
              ],
              "related_topic_ids": [
                "topic_vectors",
                "topic_vector_operations"
              ],
              "glossary_terms": [
                "vector addition",
                "scalar multiplication"
              ]
            },
            {
              "id": "precalc_ch7_s04",
              "type": "lesson",
              "title": "Magnitude and Unit Vectors",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The magnitude (or length) of a vector v = <a, b> is |v| = sqrt(a^2 + b^2). This is just the distance formula applied to the vector's components. For v = <3, 4>, |v| = sqrt(9 + 16) = 5."
                },
                {
                  "kind": "text",
                  "text": "A unit vector is a vector with magnitude 1. It indicates direction only, with no scaling. To find the unit vector in the direction of v, divide by its magnitude: u = v / |v| = <a/|v|, b/|v|>."
                },
                {
                  "kind": "text",
                  "text": "Two special unit vectors have their own names: i = <1, 0> (pointing in the positive x-direction) and j = <0, 1> (pointing in the positive y-direction). Any vector <a, b> can be written as a\u00b7i + b\u00b7j. For example, <3, -2> = 3i - 2j."
                },
                {
                  "kind": "example",
                  "text": "Find the unit vector in the direction of <-5, 12>. Magnitude: sqrt(25 + 144) = sqrt(169) = 13. Unit vector: <-5/13, 12/13>."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "unit_vector_i_j"
                }
              ],
              "related_topic_ids": [
                "topic_vectors",
                "topic_vector_operations"
              ],
              "glossary_terms": [
                "magnitude",
                "unit vector"
              ]
            },
            {
              "id": "precalc_ch7_s05",
              "type": "lesson",
              "title": "The Dot Product",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "The dot product (also called the scalar product) takes two vectors and produces a single number. For u = <a, b> and v = <c, d>, the dot product is u \u00b7 v = ac + bd. Multiply corresponding components and add."
                },
                {
                  "kind": "text",
                  "text": "The dot product has a beautiful geometric meaning: u \u00b7 v = |u| \u00b7 |v| \u00b7 cos(theta), where theta is the angle between the two vectors. This lets you find the angle between vectors: cos(theta) = (u \u00b7 v) / (|u| \u00b7 |v|)."
                },
                {
                  "kind": "text",
                  "text": "Two vectors are perpendicular (orthogonal) when the angle between them is 90 degrees, which means cos(90\u00b0) = 0. So u and v are perpendicular if and only if u \u00b7 v = 0. This is a quick test for perpendicularity."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "dot_product_angle_between_vectors"
                }
              ],
              "related_topic_ids": [
                "topic_vectors",
                "topic_dot_product"
              ],
              "glossary_terms": [
                "dot product",
                "orthogonal",
                "perpendicular"
              ]
            },
            {
              "id": "precalc_ch7_s06",
              "type": "lesson",
              "title": "The Dot Product",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "u = <2, 3> and v = <6, -4>. Dot product: 2(6) + 3(-4) = 12 - 12 = 0. Since the dot product is zero, u and v are perpendicular."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_vectors",
                "topic_dot_product"
              ],
              "glossary_terms": [
                "dot product",
                "orthogonal",
                "perpendicular"
              ]
            },
            {
              "id": "precalc_ch7_s07",
              "type": "worked_example",
              "title": "Worked Example: Finding the Angle Between Vectors",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Find the angle between u = <1, 3> and v = <4, 2>."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Compute the dot product. u \u00b7 v = 1(4) + 3(2) = 4 + 6 = 10."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Find the magnitudes. |u| = sqrt(1 + 9) = sqrt(10). |v| = sqrt(16 + 4) = sqrt(20) = 2\u00b7sqrt(5)."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Use the angle formula. cos(theta) = 10 / (sqrt(10) \u00b7 2\u00b7sqrt(5)) = 10 / (2\u00b7sqrt(50)) = 10 / (2 \u00b7 5\u00b7sqrt(2)) = 10 / (10\u00b7sqrt(2)) = 1/sqrt(2)."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "angle_between_two_vectors"
                }
              ],
              "related_topic_ids": [
                "topic_vectors",
                "topic_dot_product"
              ],
              "glossary_terms": [
                "dot product",
                "magnitude"
              ]
            },
            {
              "id": "precalc_ch7_s08",
              "type": "worked_example",
              "title": "Worked Example: Finding the Angle Between Vectors",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Solve for theta. cos(theta) = 1/sqrt(2) = sqrt(2)/2. So theta = pi/4 = 45 degrees. The vectors are 45 degrees apart."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_vectors",
                "topic_dot_product"
              ],
              "glossary_terms": [
                "dot product",
                "magnitude"
              ]
            },
            {
              "id": "precalc_ch7_s09",
              "type": "lesson",
              "title": "Introduction to Parametric Equations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Normally we describe a curve as y = f(x). But some curves are hard or impossible to write this way \u2014 for example, a circle (which fails the vertical line test) or the path of a projectile where we also want to track time."
                },
                {
                  "kind": "text",
                  "text": "Parametric equations solve this by expressing both x and y as functions of a third variable, usually t (often representing time): x = f(t) and y = g(t). As t varies over some interval, the point (x, y) traces out a curve in the plane."
                },
                {
                  "kind": "text",
                  "text": "The parameter t adds extra information: it tells you not just where the curve goes, but when the moving point reaches each location and in what direction it travels. This makes parametric equations ideal for describing motion."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "parametric_unit_circle_trace"
                }
              ],
              "related_topic_ids": [
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "parametric equations",
                "parameter"
              ]
            },
            {
              "id": "precalc_ch7_s10",
              "type": "lesson",
              "title": "Introduction to Parametric Equations",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "x = cos(t) and y = sin(t) for t in [0, 2\u00b7pi] traces out the unit circle counterclockwise. At t = 0 the point is (1, 0), at t = pi/2 it is (0, 1), at t = pi it is (-1, 0), and so on."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "parametric equations",
                "parameter"
              ]
            },
            {
              "id": "precalc_ch7_s11",
              "type": "lesson",
              "title": "Graphing and Converting Parametric Equations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "To graph a parametric curve, make a table of t values. For each t, compute x and y, then plot the points (x, y). Connect them in order of increasing t, using arrows to show direction."
                },
                {
                  "kind": "text",
                  "text": "Sometimes you can eliminate the parameter to get a Cartesian equation. For x = t + 1 and y = t^2, solve the first for t: t = x - 1. Substitute into the second: y = (x - 1)^2. This is a parabola. But note that t might be restricted (say t >= 0), which restricts what part of the parabola is traced."
                },
                {
                  "kind": "text",
                  "text": "Elimination is useful for understanding the shape, but the parametric form often carries more information \u2014 direction of travel, speed, and timing. In calculus, you will compute derivatives and integrals with respect to the parameter directly."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "parametric_to_cartesian_ellipse"
                }
              ],
              "related_topic_ids": [
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "parametric equations",
                "eliminate the parameter",
                "Cartesian equation"
              ]
            },
            {
              "id": "precalc_ch7_s12",
              "type": "lesson",
              "title": "Graphing and Converting Parametric Equations",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "x = 3\u00b7cos(t), y = 2\u00b7sin(t). Eliminate t: cos(t) = x/3 and sin(t) = y/2. Using cos^2 + sin^2 = 1: x^2/9 + y^2/4 = 1. This is an ellipse."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "parametric equations",
                "eliminate the parameter",
                "Cartesian equation"
              ]
            },
            {
              "id": "precalc_ch7_s13",
              "type": "worked_example",
              "title": "Worked Example: Parametric Equations of a Projectile",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "A ball is launched at 30 m/s at an angle of 60\u00b0 above the horizontal. Write parametric equations for its position (ignoring air resistance) and find how far it travels horizontally before hitting the ground."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Find initial velocity components. v_x = 30\u00b7cos(60\u00b0) = 30\u00b7(1/2) = 15 m/s. v_y = 30\u00b7sin(60\u00b0) = 30\u00b7(sqrt(3)/2) = 15\u00b7sqrt(3) \u2248 25.98 m/s."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Write parametric equations. x(t) = 15t (constant horizontal velocity, no air resistance). y(t) = 15\u00b7sqrt(3)\u00b7t - (1/2)(9.8)t^2 = 15\u00b7sqrt(3)\u00b7t - 4.9t^2."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "projectile_parametric_trajectory"
                }
              ],
              "related_topic_ids": [
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "parametric equations",
                "projectile motion"
              ]
            },
            {
              "id": "precalc_ch7_s14",
              "type": "worked_example",
              "title": "Worked Example: Parametric Equations of a Projectile",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 3: Find when the ball hits the ground. Set y = 0: t(15\u00b7sqrt(3) - 4.9t) = 0. So t = 0 (launch) or t = 15\u00b7sqrt(3)/4.9 \u2248 25.98/4.9 \u2248 5.30 seconds."
                },
                {
                  "kind": "step",
                  "text": "Step 4: Find horizontal distance. x(5.30) = 15 \u00b7 5.30 \u2248 79.5 meters. The ball lands about 79.5 meters away."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "parametric equations",
                "projectile motion"
              ]
            },
            {
              "id": "precalc_ch7_s15",
              "type": "lesson",
              "title": "Vectors and Parametric Equations Together",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Vectors and parametric equations are closely related. A vector-valued function r(t) = <x(t), y(t)> assigns a position vector to each value of t. As t changes, the tip of the vector traces out a curve \u2014 this is exactly a parametric curve."
                },
                {
                  "kind": "text",
                  "text": "The velocity vector is the derivative of the position vector with respect to t: v(t) = <x'(t), y'(t)>. Its direction tells which way the object is moving, and its magnitude |v(t)| gives the speed. This is a preview of how calculus handles motion in the plane."
                },
                {
                  "kind": "text",
                  "text": "A simple but important example: a line through point P = (x0, y0) in the direction of vector d = <a, b> can be parametrized as r(t) = <x0 + at, y0 + bt>. Here t = 0 gives the starting point, and increasing t moves you along the line in the direction of d."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "vector_valued_function_curve"
                }
              ],
              "related_topic_ids": [
                "topic_vectors",
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "vector-valued function",
                "position vector",
                "velocity vector"
              ]
            },
            {
              "id": "precalc_ch7_s16",
              "type": "lesson",
              "title": "Vectors and Parametric Equations Together",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "The line through (2, -1) in the direction <3, 4> has parametric equations x = 2 + 3t, y = -1 + 4t. At t = 0: (2, -1). At t = 1: (5, 3). At t = -1: (-1, -5)."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_vectors",
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "vector-valued function",
                "position vector",
                "velocity vector"
              ]
            },
            {
              "id": "precalc_ch7_s17",
              "type": "practice",
              "title": "Practice: Vectors and Parametric Equations",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) Given u = <-3, 7> and v = <2, 4>, find u + v, u - v, 2u, and u \u00b7 v."
                },
                {
                  "kind": "problem",
                  "text": "2) Find the unit vector in the direction of <5, -12>."
                },
                {
                  "kind": "problem",
                  "text": "3) Eliminate the parameter: x = t^2, y = 2t + 1. What curve does this describe?"
                },
                {
                  "kind": "problem",
                  "text": "4) Write parametric equations for an ellipse centered at the origin with semi-axes a = 4 and b = 2."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_vectors",
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "vector",
                "unit vector",
                "dot product",
                "parametric equations"
              ]
            },
            {
              "id": "precalc_ch7_s18",
              "type": "lesson",
              "title": "Why Vectors and Parametrics Matter for Calculus",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "In Calculus II, you will compute arc lengths and areas using parametric curves. You will also work with polar coordinates, which describe curves using angles and distances \u2014 a natural fit for parametric representation."
                },
                {
                  "kind": "text",
                  "text": "In Calculus III (multivariable), vectors become central. You will use vector-valued functions to describe curves in 3D space, compute tangent and normal vectors, and study the motion of objects along space curves. Concepts like the dot product extend to 3D and lead to the cross product."
                },
                {
                  "kind": "text",
                  "text": "Line integrals, surface integrals, and the major theorems of vector calculus (Green's, Stokes', Divergence) all depend on parametric descriptions of curves and surfaces. The foundations you are building now will be used throughout advanced math."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "vectors_calculus_connection"
                }
              ],
              "related_topic_ids": [
                "topic_vectors",
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "vector",
                "parametric equations",
                "line integral",
                "vector-valued function"
              ]
            },
            {
              "id": "precalc_ch7_s19",
              "type": "lesson",
              "title": "Why Vectors and Parametrics Matter for Calculus",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "If vectors feel abstract now, remember that every physical quantity involving direction \u2014 wind, ocean currents, gravity, electric fields \u2014 is best described with vectors. Learning vectors now is investing in a language you will use constantly."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_vectors",
                "topic_parametric_equations"
              ],
              "glossary_terms": [
                "vector",
                "parametric equations",
                "line integral",
                "vector-valued function"
              ]
            },
            {
              "id": "precalc_ch7_s20",
              "type": "summary",
              "title": "Chapter Summary: Vectors and Parametric Equations",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Vectors have magnitude and direction. They are written in component form <a, b> and support addition, scalar multiplication, and the dot product. The dot product u \u00b7 v = |u||v|cos(theta) connects algebra and geometry and tests for perpendicularity."
                },
                {
                  "kind": "text",
                  "text": "Parametric equations x = f(t), y = g(t) describe curves by expressing both coordinates as functions of a parameter. They handle curves that are not functions, describe motion with timing, and can be converted to Cartesian form by eliminating the parameter."
                },
                {
                  "kind": "text",
                  "text": "Together, vectors and parametric equations form the language of motion and multidimensional geometry. They are essential for studying curves, surfaces, and physical quantities in Calculus II and III."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "vectors_parametric_summary"
                }
              ],
              "related_topic_ids": [
                "topic_vectors",
                "topic_parametric_equations",
                "topic_dot_product"
              ],
              "glossary_terms": [
                "vector",
                "magnitude",
                "dot product",
                "parametric equations",
                "parameter",
                "unit vector"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch7_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Vectors and Parametric Equations",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch7_test",
            "optional_recommended": true,
            "title": "Chapter Test: Vectors and Parametric Equations",
            "questions": []
          }
        },
        {
          "id": "precalc_ch8_intro_limits",
          "title": "Introduction to Limits",
          "description": "Get a gentle first look at limits \u2014 the concept that launches all of calculus. Build intuition through tables, graphs, and simple examples before the formal treatment in Calculus I.",
          "micro_quiz_interval": 5,
          "slides": [
            {
              "id": "precalc_ch8_s01",
              "type": "lesson",
              "title": "The Big Idea: What Is a Limit?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Imagine walking toward a wall. You get closer and closer, but you never have to actually touch it. A limit works the same way: it describes the value a function approaches as the input gets closer and closer to some number."
                },
                {
                  "kind": "text",
                  "text": "We write lim(x -> a) f(x) = L to mean: as x gets closer to a (from both sides), f(x) gets closer to L. The function does not need to equal L at x = a. It does not even need to be defined at x = a. All that matters is the behavior near a."
                },
                {
                  "kind": "text",
                  "text": "Limits answer the question: what value is f(x) heading toward? This question turns out to be the foundation of all of calculus. Derivatives are defined using limits. Integrals are defined using limits. Understanding limits is understanding the engine that drives calculus."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "limit_approaching_hole"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro"
              ],
              "glossary_terms": [
                "limit"
              ]
            },
            {
              "id": "precalc_ch8_s02",
              "type": "lesson",
              "title": "The Big Idea: What Is a Limit?",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Consider f(x) = (x^2 - 1)/(x - 1). At x = 1, this is 0/0 \u2014 undefined. But simplify: (x-1)(x+1)/(x-1) = x + 1 for x \u2260 1. As x approaches 1, x + 1 approaches 2. So lim(x -> 1) f(x) = 2, even though f(1) does not exist."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro"
              ],
              "glossary_terms": [
                "limit"
              ]
            },
            {
              "id": "precalc_ch8_s03",
              "type": "lesson",
              "title": "Estimating Limits from Tables",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "One way to find a limit is to make a table of values. Choose x-values that approach a from both sides and compute f(x) for each. If the outputs approach the same number from both directions, that number is the limit."
                },
                {
                  "kind": "text",
                  "text": "For f(x) = (x^2 - 4)/(x - 2), approach x = 2 from the left: x = 1.9 gives f = 3.9, x = 1.99 gives f = 3.99, x = 1.999 gives f = 3.999. From the right: x = 2.1 gives f = 4.1, x = 2.01 gives f = 4.01, x = 2.001 gives f = 4.001. The outputs approach 4 from both sides."
                },
                {
                  "kind": "text",
                  "text": "So lim(x -> 2) (x^2 - 4)/(x - 2) = 4. We can verify algebraically: (x^2 - 4)/(x - 2) = (x + 2)(x - 2)/(x - 2) = x + 2 for x \u2260 2. Plug in x = 2: 2 + 2 = 4."
                }
              ],
              "graphics": [
                {
                  "kind": "table",
                  "name": "limit_estimation_table"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro"
              ],
              "glossary_terms": [
                "limit"
              ]
            },
            {
              "id": "precalc_ch8_s04",
              "type": "lesson",
              "title": "Estimating Limits from Tables",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "Tables give numerical evidence but are not rigorous proof. They are a great way to build intuition, especially when the algebra is hard to simplify."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro"
              ],
              "glossary_terms": [
                "limit"
              ]
            },
            {
              "id": "precalc_ch8_s05",
              "type": "lesson",
              "title": "Estimating Limits from Graphs",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "You can also estimate a limit by looking at a graph. Trace along the curve from the left side, approaching x = a. What y-value does the curve head toward? Then do the same from the right side. If both sides approach the same y-value, that is the limit."
                },
                {
                  "kind": "text",
                  "text": "If there is a hole in the graph at x = a (an open circle), the limit still exists \u2014 it is the y-value of the hole. The function might not be defined there, but the limit describes where the curve was heading."
                },
                {
                  "kind": "text",
                  "text": "If the curve approaches different y-values from the left and right, the limit does not exist. For example, a step function that jumps from 1 to 3 at x = 2 has a left-hand limit of 1 and a right-hand limit of 3. Since they disagree, lim(x -> 2) does not exist."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "limit_from_graph_hole_jump"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_one_sided_limits"
              ],
              "glossary_terms": [
                "limit",
                "one-sided limit",
                "does not exist"
              ]
            },
            {
              "id": "precalc_ch8_s06",
              "type": "lesson",
              "title": "Estimating Limits from Graphs",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Looking at the graph of f(x) = |x|/x: from the left of 0, the function equals -1. From the right, it equals 1. The left and right limits disagree, so lim(x -> 0) |x|/x does not exist."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_one_sided_limits"
              ],
              "glossary_terms": [
                "limit",
                "one-sided limit",
                "does not exist"
              ]
            },
            {
              "id": "precalc_ch8_s07",
              "type": "lesson",
              "title": "One-Sided Limits",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A one-sided limit looks at the function's behavior from only one direction. The left-hand limit lim(x -> a^-) f(x) examines what happens as x approaches a from values less than a. The right-hand limit lim(x -> a^+) f(x) examines approach from values greater than a."
                },
                {
                  "kind": "text",
                  "text": "The two-sided limit lim(x -> a) f(x) exists if and only if both one-sided limits exist and are equal. When they differ, the two-sided limit does not exist, but each one-sided limit might still be a perfectly good number."
                },
                {
                  "kind": "text",
                  "text": "One-sided limits are especially important for piecewise functions, which have different formulas on different intervals. At the boundary between pieces, the left-hand and right-hand limits may differ."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "one_sided_limits_piecewise"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_one_sided_limits"
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              "glossary_terms": [
                "one-sided limit",
                "left-hand limit",
                "right-hand limit",
                "piecewise function"
              ]
            },
            {
              "id": "precalc_ch8_s08",
              "type": "lesson",
              "title": "One-Sided Limits",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "Let f(x) = {x + 1 if x < 3, 2x - 2 if x >= 3}. Left-hand limit at 3: lim(x -> 3^-) (x + 1) = 4. Right-hand limit at 3: lim(x -> 3^+) (2x - 2) = 4. Both equal 4, so lim(x -> 3) f(x) = 4."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_one_sided_limits"
              ],
              "glossary_terms": [
                "one-sided limit",
                "left-hand limit",
                "right-hand limit",
                "piecewise function"
              ]
            },
            {
              "id": "precalc_ch8_s09",
              "type": "worked_example",
              "title": "Worked Example: Finding a Limit Algebraically",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Find lim(x -> 3) (x^2 - 9)/(x - 3)."
                },
                {
                  "kind": "step",
                  "text": "Step 1: Try direct substitution. Plugging in x = 3: (9 - 9)/(3 - 3) = 0/0. This is an indeterminate form, so we need to simplify."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Factor the numerator. x^2 - 9 = (x - 3)(x + 3)."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Cancel the common factor. (x - 3)(x + 3)/(x - 3) = x + 3 for x \u2260 3."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "limit_hole_at_3"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_indeterminate_forms"
              ],
              "glossary_terms": [
                "limit",
                "indeterminate form",
                "factor and cancel"
              ]
            },
            {
              "id": "precalc_ch8_s10",
              "type": "worked_example",
              "title": "Worked Example: Finding a Limit Algebraically",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Now substitute x = 3 into the simplified expression: 3 + 3 = 6. So the limit is 6."
                },
                {
                  "kind": "note",
                  "text": "The 0/0 indeterminate form is a signal to simplify, not a final answer. Factor, cancel, and try substitution again. This pattern will appear often in calculus."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_indeterminate_forms"
              ],
              "glossary_terms": [
                "limit",
                "indeterminate form",
                "factor and cancel"
              ]
            },
            {
              "id": "precalc_ch8_s11",
              "type": "lesson",
              "title": "Limits That Go to Infinity",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Sometimes as x approaches a value, f(x) grows without bound \u2014 it shoots up or down toward infinity. In this case, we say the limit is positive infinity or negative infinity. Technically, the limit does not exist as a finite number, but writing lim = infinity tells us how the function behaves."
                },
                {
                  "kind": "text",
                  "text": "For example, lim(x -> 0^+) 1/x = +infinity: as x approaches 0 from the right, 1/x gets larger and larger without bound. And lim(x -> 0^-) 1/x = -infinity: from the left, 1/x becomes more and more negative."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "limit_to_infinity_1_over_x"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_infinite_limits"
              ],
              "glossary_terms": [
                "infinite limit",
                "limit at infinity",
                "horizontal asymptote"
              ]
            },
            {
              "id": "precalc_ch8_s12",
              "type": "lesson",
              "title": "Limits That Go to Infinity",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "We can also ask what happens as x itself goes to infinity. lim(x -> infinity) 1/x = 0: as x gets huge, 1/x gets closer and closer to 0. This is called a limit at infinity, and it describes the long-run behavior of the function \u2014 closely related to horizontal asymptotes."
                },
                {
                  "kind": "example",
                  "text": "lim(x -> infinity) (3x + 1)/(x - 2). Divide numerator and denominator by x: (3 + 1/x)/(1 - 2/x). As x -> infinity, 1/x -> 0 and 2/x -> 0, so the limit is 3/1 = 3. Horizontal asymptote y = 3."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_infinite_limits"
              ],
              "glossary_terms": [
                "infinite limit",
                "limit at infinity",
                "horizontal asymptote"
              ]
            },
            {
              "id": "precalc_ch8_s13",
              "type": "lesson",
              "title": "The Famous Limit: sin(x)/x",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "One of the most important limits in all of calculus is lim(x -> 0) sin(x)/x = 1, where x is measured in radians. Direct substitution gives 0/0, so you cannot just plug in. But numerical and graphical evidence shows the ratio approaches 1."
                },
                {
                  "kind": "text",
                  "text": "Make a table: x = 0.1 gives sin(0.1)/0.1 \u2248 0.9983. x = 0.01 gives sin(0.01)/0.01 \u2248 0.99998. x = 0.001 gives sin(0.001)/0.001 \u2248 0.9999998. Closer and closer to 1 from below."
                },
                {
                  "kind": "text",
                  "text": "Why does this matter? This limit is used to derive the derivatives of sin(x) and cos(x) in Calculus I. It is also why we use radians \u2014 in degrees, sin(x)/x would approach pi/180 instead of 1, making all the formulas messy."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "sinx_over_x_graph"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_trig_limits"
              ],
              "glossary_terms": [
                "limit",
                "radian"
              ]
            },
            {
              "id": "precalc_ch8_s14",
              "type": "lesson",
              "title": "The Famous Limit: sin(x)/x",
              "content_blocks": [
                {
                  "kind": "note",
                  "text": "A companion limit is lim(x -> 0) (1 - cos(x))/x = 0. Both of these are worth remembering because they appear frequently in the derivation of trig derivatives."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_trig_limits"
              ],
              "glossary_terms": [
                "limit",
                "radian"
              ]
            },
            {
              "id": "precalc_ch8_s15",
              "type": "lesson",
              "title": "What Does Continuity Mean?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A function is continuous at a point x = a if three conditions are met: (1) f(a) is defined, (2) lim(x -> a) f(x) exists, and (3) lim(x -> a) f(x) = f(a). In plain language, the function has no holes, jumps, or breaks at that point."
                },
                {
                  "kind": "text",
                  "text": "If you can draw the graph through x = a without lifting your pencil, the function is continuous there. Most common functions \u2014 polynomials, trig functions, exponentials \u2014 are continuous on their entire domains."
                },
                {
                  "kind": "text",
                  "text": "Discontinuities come in three types. A removable discontinuity (hole) is where the limit exists but the function is not defined or has the wrong value. A jump discontinuity is where the left and right limits differ. An infinite discontinuity is where the function blows up to infinity (like at a vertical asymptote)."
                }
              ],
              "graphics": [
                {
                  "kind": "diagram",
                  "name": "three_types_discontinuity"
                }
              ],
              "related_topic_ids": [
                "topic_continuity"
              ],
              "glossary_terms": [
                "continuous",
                "discontinuity",
                "removable discontinuity",
                "jump discontinuity",
                "infinite discontinuity"
              ]
            },
            {
              "id": "precalc_ch8_s16",
              "type": "lesson",
              "title": "What Does Continuity Mean?",
              "content_blocks": [
                {
                  "kind": "example",
                  "text": "f(x) = (x^2 - 1)/(x - 1) has a removable discontinuity at x = 1 (a hole). g(x) = 1/x has an infinite discontinuity at x = 0 (a vertical asymptote). A step function has a jump discontinuity."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_continuity"
              ],
              "glossary_terms": [
                "continuous",
                "discontinuity",
                "removable discontinuity",
                "jump discontinuity",
                "infinite discontinuity"
              ]
            },
            {
              "id": "precalc_ch8_s17",
              "type": "worked_example",
              "title": "Worked Example: Checking Continuity of a Piecewise Function",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "Is the function f(x) = {x^2 if x < 2, 3x - 2 if x >= 2} continuous at x = 2?"
                },
                {
                  "kind": "step",
                  "text": "Step 1: Check that f(2) is defined. Since x = 2 falls in the second piece: f(2) = 3(2) - 2 = 4."
                },
                {
                  "kind": "step",
                  "text": "Step 2: Find the left-hand limit. lim(x -> 2^-) x^2 = 4."
                },
                {
                  "kind": "step",
                  "text": "Step 3: Find the right-hand limit. lim(x -> 2^+) (3x - 2) = 4."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "piecewise_continuity_check"
                }
              ],
              "related_topic_ids": [
                "topic_continuity",
                "topic_one_sided_limits"
              ],
              "glossary_terms": [
                "continuous",
                "one-sided limit",
                "piecewise function"
              ]
            },
            {
              "id": "precalc_ch8_s18",
              "type": "worked_example",
              "title": "Worked Example: Checking Continuity of a Piecewise Function",
              "content_blocks": [
                {
                  "kind": "step",
                  "text": "Step 4: Compare. Left limit = right limit = 4, and f(2) = 4. All three match, so f is continuous at x = 2."
                },
                {
                  "kind": "note",
                  "text": "If the left and right limits had disagreed, or if they matched but did not equal f(2), the function would be discontinuous at x = 2."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_continuity",
                "topic_one_sided_limits"
              ],
              "glossary_terms": [
                "continuous",
                "one-sided limit",
                "piecewise function"
              ]
            },
            {
              "id": "precalc_ch8_s19",
              "type": "lesson",
              "title": "Limits and the Road to Calculus",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Everything in calculus is built on limits. The derivative \u2014 measuring instantaneous rate of change \u2014 is defined as a limit: f'(a) = lim(h -> 0) (f(a+h) - f(a))/h. Without limits, there would be no way to make 'instantaneous' precise."
                },
                {
                  "kind": "text",
                  "text": "The definite integral \u2014 computing areas and accumulated quantities \u2014 is also defined as a limit: the limit of Riemann sums as the number of rectangles approaches infinity. Without limits, there would be no way to make 'infinitely many infinitely thin slices' rigorous."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "limits_foundation_of_calculus"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro"
              ],
              "glossary_terms": [
                "limit",
                "derivative",
                "integral",
                "Riemann sum"
              ]
            },
            {
              "id": "precalc_ch8_s20",
              "type": "lesson",
              "title": "Limits and the Road to Calculus",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "By understanding limits intuitively now, you are preparing for the two pillars of calculus. In Calculus I, you will learn precise limit laws, the formal definition, and how limits connect to derivatives and integrals. But the core idea is what you have already learned: what value is the function approaching?"
                },
                {
                  "kind": "note",
                  "text": "If you remember one thing from this chapter, remember this: limits describe the destination, not the journey. The function does not need to reach the value \u2014 it just needs to be heading there."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro"
              ],
              "glossary_terms": [
                "limit",
                "derivative",
                "integral",
                "Riemann sum"
              ]
            },
            {
              "id": "precalc_ch8_s21",
              "type": "practice",
              "title": "Practice: Introduction to Limits",
              "content_blocks": [
                {
                  "kind": "problem",
                  "text": "1) Evaluate lim(x -> 4) (x^2 - 16)/(x - 4) by simplifying algebraically."
                },
                {
                  "kind": "problem",
                  "text": "2) For the piecewise function f(x) = {2x + 1 if x < 1, 5 if x = 1, x^2 + 2 if x > 1}, find lim(x -> 1^-) f(x), lim(x -> 1^+) f(x), and determine if lim(x -> 1) f(x) exists."
                },
                {
                  "kind": "problem",
                  "text": "3) Evaluate lim(x -> infinity) (5x^2 - 3)/(2x^2 + x)."
                },
                {
                  "kind": "problem",
                  "text": "4) Is f(x) = {x + 3 if x \u2260 2, 10 if x = 2} continuous at x = 2? Explain why or why not."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro",
                "topic_continuity"
              ],
              "glossary_terms": [
                "limit",
                "one-sided limit",
                "continuous"
              ]
            },
            {
              "id": "precalc_ch8_s22",
              "type": "summary",
              "title": "Chapter Summary: Introduction to Limits",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A limit describes the value a function approaches as its input gets closer to some target. We write lim(x -> a) f(x) = L. The function does not need to equal L at x = a \u2014 limits are about the trend, not the exact value."
                },
                {
                  "kind": "text",
                  "text": "Limits can be estimated using tables and graphs. One-sided limits examine approach from the left or right alone. The two-sided limit exists only if both one-sided limits agree. Common algebraic techniques include factoring and canceling."
                },
                {
                  "kind": "text",
                  "text": "Limits can be infinite (function blows up) or evaluated at infinity (long-run behavior). The special limit sin(x)/x -> 1 as x -> 0 is foundational for trig in calculus."
                }
              ],
              "graphics": [
                {
                  "kind": "concept",
                  "name": "limits_chapter_summary"
                }
              ],
              "related_topic_ids": [
                "topic_limits_intro",
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              "glossary_terms": [
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                "continuous",
                "discontinuity",
                "indeterminate form",
                "infinite limit"
              ]
            },
            {
              "id": "precalc_ch8_s23",
              "type": "summary",
              "title": "Chapter Summary: Introduction to Limits",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "Continuity means no holes, jumps, or asymptotes: lim f(x) = f(a). Limits are the engine of calculus \u2014 derivatives and integrals are both defined as limits. You are now ready for Calculus I."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "topic_limits_intro",
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                "topic_continuity"
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              "glossary_terms": [
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                "continuous",
                "discontinuity",
                "indeterminate form",
                "infinite limit"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "precalc_ch8_midquiz",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz: Introduction to Limits",
            "questions": []
          },
          "final_test": {
            "id": "precalc_ch8_test",
            "optional_recommended": true,
            "title": "Chapter Test: Introduction to Limits",
            "questions": []
          }
        }
      ]
    },
    {
      "id": "calc_1",
      "title": "Calc 1 Pathway",
      "level": "Calculus I",
      "description": "Limits, derivatives, and first applications.",
      "chapters": [
        {
          "id": "calc1_limits",
          "title": "Limits Fundamentals",
          "description": "Core ideas for approaching values.",
          "micro_quiz_interval": 4,
          "slides": [
            {
              "id": "s_limit_intro",
              "type": "lesson",
              "title": "What is a limit?",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "A limit describes behavior near a point, not only at the point."
                }
              ],
              "graphics": [
                {
                  "kind": "graph",
                  "name": "approach-point"
                }
              ],
              "related_topic_ids": [
                "concept_limits_intro"
              ],
              "glossary_terms": [
                "limit",
                "continuity"
              ]
            },
            {
              "id": "s_limit_infinity",
              "type": "lesson",
              "title": "Limits at infinity",
              "content_blocks": [
                {
                  "kind": "text",
                  "text": "As x grows very large, some expressions settle toward finite values."
                }
              ],
              "graphics": [],
              "related_topic_ids": [
                "concept_limit_at_infinity"
              ],
              "glossary_terms": [
                "infinity",
                "asymptote"
              ]
            }
          ],
          "midpoint_quiz": {
            "id": "q_calc1_limits_mid",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz",
            "questions": [
              {
                "id": "q1",
                "type": "multiple_choice",
                "prompt": "lim(x->infinity) 1/x equals:",
                "choices": [
                  "0",
                  "1",
                  "infinity",
                  "-1"
                ],
                "correct_choice_index": 0,
                "explanation": "The denominator grows without bound so the fraction shrinks toward 0."
              }
            ]
          },
          "final_test": {
            "id": "t_calc1_limits_final",
            "optional_recommended": true,
            "title": "Chapter Test",
            "questions": []
          }
        }
      ]
    },
    {
      "id": "calc_2",
      "title": "Calc 2 Pathway",
      "level": "Calculus II",
      "description": "Integrals, techniques, and sequences/series.",
      "chapters": [
        {
          "id": "calc2_integrals",
          "title": "Integral Techniques",
          "description": "Build antiderivative intuition and computation flow.",
          "micro_quiz_interval": 5,
          "slides": [],
          "midpoint_quiz": {
            "id": "q_calc2_integrals_mid",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz",
            "questions": []
          },
          "final_test": {
            "id": "t_calc2_integrals_final",
            "optional_recommended": true,
            "title": "Chapter Test",
            "questions": []
          }
        }
      ]
    },
    {
      "id": "advanced_calc",
      "title": "Advanced Calculus Pathway",
      "level": "Advanced",
      "description": "Higher-dimensional and theoretical calculus extensions.",
      "chapters": [
        {
          "id": "adv_multivariable_intro",
          "title": "Multivariable Foundations",
          "description": "Partial derivatives and gradient intuition.",
          "micro_quiz_interval": 6,
          "slides": [],
          "midpoint_quiz": {
            "id": "q_adv_multi_mid",
            "required_to_take": true,
            "required_to_pass": false,
            "title": "Mid-Chapter Quiz",
            "questions": []
          },
          "final_test": {
            "id": "t_adv_multi_final",
            "optional_recommended": true,
            "title": "Chapter Test",
            "questions": []
          }
        }
      ]
    }
  ]
}