# Parallelogram

Given: |AB| = 76cm, |BC| = 44cm, angle BAD = 30°.

Find the area of the parallelogram.

To solve this problem, we need to know the following concepts:
- The formula to calculate the area of a parallelogram is base * height.
- We can use trigonometric functions to find the height of the parallelogram.

Related problems:
- Right angle: If a, b, and c are sides of a triangle ABC with a right angle in A, find the value of each missing side. Given b = 10, c = 6.
- Triangle: Calculate the area of a triangle with two sides long 7 and 19 and an included angle of 36°.
- Rhomboid: Calculate the circumference and area of a rhomboid with sides 20 and 14 and an angle of 50°.
- Isosceles right triangle: Calculate the length of the base of an isosceles right triangle with an area of 18 dm^2.
- Triangle: Calculate the area of a right triangle ΔABC with one leg long 14 and its opposite angle 59°.
- The cable car: Calculate the altitude difference between the upper and lower stations of a cable car with a length of 3.5 kilometers and an angle of climb of 30 degrees.
- The cable car: Calculate the height difference between the lower and upper stations of a cable car that is 2610 m long and rises at an angle of 35°.
- Cable car 2: Calculate the length of the track of a cable car that rises at an angle of 41° and connects the upper and lower station with an altitude difference of 1175 m.
- 30-60-90 triangle: Calculate the length of the shorter leg of a 30°-60°-90° triangle with a longer leg of 5.
- Angle: Calculate angle x in a triangle ABC where side AB is 10, and side AC is 15 (hypotenuse).
- Obtuse angle: Calculate the size of angle DMO in a triangle DOM where line MN is the bisector of angle DMO and the obtuse angle between lines MN and OH is four times larger than angle DMN.
- Climb: Calculate the road pitch in ‰ (permille) for a horizontal distance of 4.2 km and a road rise of 6.3 m.
- Right triangle: Calculate the other two angles or the length of the leg of the yard for a right triangle with a ladder of 16 feet reaching up 14 feet on a house wall.
- Right triangle: Calculate the length of the remaining sides of a right triangle ABC with side a = 19 and area S = 95.
- Height: Calculate the height of an equilateral triangle with a side of 38.
- The ladder: Calculate the height reached by a ladder with a length of 3 m leaning against a wall at an inclination of 45°.
- Center traverse: Determine if the middle traverse bisects the triangle.


To calculate dew point, you need to know the current temperature and relative humidity. The equation to solve is Td = T - ((100 - RH) / 5), where Td is the dew point temperature in degrees Celsius, T is the observed temperature in degrees Celsius, and RH is the relative humidity in percent. This equation is accurate for humidity values above 50 percent.

Relative humidity is the ratio of how much moisture the air is holding to how much moisture it could hold at a given temperature. To use the equation, put the values for the temperature and relative humidity into it. For example, with an observed temperature of 19 degrees Celsius and a relative humidity of 75 percent, the equation becomes: Td = 19 - ((100 - 75) / 5).

Solving the equation yields the dew point temperature. Continuing the example: Td = 19 - ((100 - 75) / 5) = 19 - (25 / 5) = 19 - 5 = 14. This means the dew point temperature is 14 degrees Celsius.


# Mathematical Equations to Solve

Mathematical equations to solve can help students understand the material and improve their grades. Solving math problems is essential for academic success.

## Key Concepts in Mathematics

To develop mathematical literacy, it's crucial to connect mathematics to real-life situations. This can be achieved by highlighting the cultivation of students' ability to understand, analyze, operate, explore, express, and apply knowledge to solve problems. The key points of knowledge include division of the remainder, recognition of numbers, decimeters and millimeters, addition, recognition of directions, and other fundamental concepts.

## Mathematical Tools and Resources

A personal income tax calculator can be a useful tool for students to learn about real-world applications of mathematics. The calculator can be accessed through the Baidu app, where users can enter information such as pre-tax income to calculate after-tax income. Scientific calculators also play a vital role in learning algebra, trigonometry, geometry, calculus, and other mathematical disciplines.

## Effective Math Solvers

A reliable math solver can provide step-by-step solutions to mathematical problems, along with explanations to help students understand the underlying concepts. Such tools can be invaluable for students, enabling them to learn new things and develop a deeper understanding of mathematics.

## Testimonials from Satisfied Users

* "This app is really good! I love that it's not just giving answers but the steps as well. And if I don't understand the steps, there are explanations!" - Damaris Rivera
* "It's an amazing app. Not only does it solve the answers for you, it shows you the work too. I've even written problems and it figured it out. The ultimate tool for students."

Note: The original text contained irrelevant information about foreign money-making websites, cross-border background volume, and clothing designs, which have been removed to refine the content and focus on mathematical equations and related concepts.


The general form of a quadratic function presents the function in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \ne 0$. If $a > 0$, the parabola opens upward. If $a < 0$, the parabola opens downward. The axis of symmetry is defined by $x = -\frac{b}{2a}$.

The standard form of a quadratic function presents the function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function. As with the general form, if $a > 0$, the parabola opens upward and the vertex is a minimum. If $a < 0$, the parabola opens downward, and the vertex is a maximum.

To find the equation of a quadratic function in general form, identify the horizontal shift of the parabola, $h$, and the vertical shift of the parabola, $k$. Substitute the values of $h$ and $k$ into the function $f(x) = a(x - h)^2 + k$. Substitute the values of any point, other than the vertex, on the graph of the parabola for $x$ and $f(x)$. Solve for the stretch factor, $|a|$. Expand and simplify to write in general form.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

To find the domain and range of a quadratic function, identify the domain as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

Given a quadratic function $f(x) = -5x^2 + 9x - 1$, the domain is all real numbers. Because $a$ is negative, the parabola opens downward and has a maximum value. The maximum value is given by $f(h)$, where $h = -\frac{b}{2a}$. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Given a quadratic function $f(x) = 2(x - \frac{4}{7})^2 + \frac{8}{11}$, the domain is all real numbers. Because $a$ is positive, the parabola opens upward and has a minimum value. The minimum value is given by $f(h)$, where $h = \frac{4}{7}$. The range is $f(x) \ge \frac{8}{11}$, or $[\frac{8}{11}, \infty)$.

To write an equation for the quadratic function $g$ in the graph, we can see the graph of $g$ is the graph of $f(x) = x^2$ shifted to the left 2 and down 3, giving a formula in the form $g(x) = a(x + 2)^2 - 3$. Substituting the coordinates of a point on the curve, such as $(0, -1)$, we can solve for the stretch factor. In standard form, the algebraic model for this graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. To write this in general polynomial form, we can expand the formula and simplify terms.

The vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$ is given by $h = -\frac{b}{2a}$ and $k = f(h)$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Given the equation $g(x) = 13 + x^2 - 6x$, we can write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

A coordinate grid has been superimposed over the quadratic path of a basketball in the picture. Find an equation for the path of the ball. Does the shooter make the basket? The equation for the path of the ball is $f(x) = -\frac{1}{2}(x - 2)^2 + 5$. The shooter does make the basket.

Find the domain and range of $f(x) = 2(x - \frac{4}{7})^2 + \frac{8}{11}$. The domain is all real numbers. The range is $f(x) \ge \frac{8}{11}$, or $[\frac{8}{11}, \infty)$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

A quadratic function is a function of degree two. The graph of a quadratic function is a parabola. The general form of a quadratic function is $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are real numbers and $a \ne 0$. The standard form of a quadratic function is $f(x) = a(x - h)^2 + k$. The vertex $(h, k)$ is located at $h = -\frac{b}{2a}$ and $k = f(h)$.

Given a graph of a quadratic function, write the equation of the function in general form. Identify the horizontal shift of the parabola; this value is $h$. Identify the vertical shift of the parabola; this value is $k$. Substitute the values of the horizontal and vertical shift for $h$ and $k$ in the function $f(x) = a(x - h)^2 + k$. Substitute the values of any point, other than the vertex, on the graph of the parabola for $x$ and $f(x)$. Solve for the stretch factor, $|a|$. Expand and simplify to write in general form.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$. Therefore, when a quadratic function is given in the form $f(x) = ax^2 + bx + c$, we can find the vertex by using the formula $h = -\frac{b}{2a}$. Then we can find $k$ by evaluating $f(h)$.

The vertex form of a quadratic function $f(x) = a(x - h)^2 + k$ is equivalent to the standard form $f(x) = ax^2 + bx + c$. We can convert from the standard form to the vertex form by completing the square. Given $f(x) = ax^2 + bx + c$, we can complete the square to write the function in vertex form.

To convert $f(x) = ax^2 + bx + c$ to vertex form, $f(x) = a(x - h)^2 + k$, we need to complete the square on the quadratic term. We can complete the square to write the function in the form $f(x) = a(x - h)^2 + k$.

Given a quadratic function, the vertex form $f(x) = a(x - h)^2 + k$ provides an easy way to determine the graph’s vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. When the function is written in vertex form, it is easy to identify the vertex because the values of $h$ and $k$ are given.

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry of the parabola is the vertical line $x = h$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$. Therefore, when a quadratic function is given in the form $f(x) = ax^2 + bx + c$, we can find the vertex by using the formula $h = -\frac{b}{2a}$. Then we can find $k$ by evaluating $f(h)$.

The vertex form of a quadratic function $f(x) = a(x - h)^2 + k$ is equivalent to the standard form $f(x) = ax^2 + bx + c$. We can convert from the standard form to the vertex form by completing the square. Given $f(x) = ax^2 + bx + c$, we can complete the square to write the function in vertex form.

To convert $f(x) = ax^2 + bx + c$ to vertex form, $f(x) = a(x - h)^2 + k$, we need to complete the square on the quadratic term. We can complete the square to write the function in the form $f(x) = a(x - h)^2 + k$.

Given a quadratic function, the vertex form $f(x) = a(x - h)^2 + k$ provides an easy way to determine the graph’s vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. When the function is written in vertex form, it is easy to identify the vertex because the values of $h$ and $k$ are given.

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry of the parabola is the vertical line $x = h$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$. Therefore, when a quadratic function is given in the form $f(x) = ax^2 + bx + c$, we can find the vertex by using the formula $h = -\frac{b}{2a}$. Then we can find $k$ by evaluating $f(h)$.

The vertex form of a quadratic function $f(x) = a(x - h)^2 + k$ is equivalent to the standard form $f(x) = ax^2 + bx + c$. We can convert from the standard form to the vertex form by completing the square. Given $f(x) = ax^2 + bx + c$, we can complete the square to write the function in vertex form.

To convert $f(x) = ax^2 + bx + c$ to vertex form, $f(x) = a(x - h)^2 + k$, we need to complete the square on the quadratic term. We can complete the square to write the function in the form $f(x) = a(x - h)^2 + k$.

Given a quadratic function, the vertex form $f(x) = a(x - h)^2 + k$ provides an easy way to determine the graph’s vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. When the function is written in vertex form, it is easy to identify the vertex because the values of $h$ and $k$ are given.

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry of the parabola is the vertical line $x = h$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$. Therefore, when a quadratic function is given in the form $f(x) = ax^2 + bx + c$, we can find the vertex by using the formula $h = -\frac{b}{2a}$. Then we can find $k$ by evaluating $f(h)$.

The vertex form of a quadratic function $f(x) = a(x - h)^2 + k$ is equivalent to the standard form $f(x) = ax^2 + bx + c$. We can convert from the standard form to the vertex form by completing the square. Given $f(x) = ax^2 + bx + c$, we can complete the square to write the function in vertex form.

To convert $f(x) = ax^2 + bx + c$ to vertex form, $f(x) = a(x - h)^2 + k$, we need to complete the square on the quadratic term. We can complete the square to write the function in the form $f(x) = a(x - h)^2 + k$.

Given a quadratic function, the vertex form $f(x) = a(x - h)^2 + k$ provides an easy way to determine the graph’s vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. When the function is written in vertex form, it is easy to identify the vertex because the values of $h$ and $k$ are given.

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry of the parabola is the vertical line $x = h$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$. Therefore, when a quadratic function is given in the form $f(x) = ax^2 + bx + c$, we can find the vertex by using the formula $h = -\frac{b}{2a}$. Then we can find $k$ by evaluating $f(h)$.

The vertex form of a quadratic function $f(x) = a(x - h)^2 + k$ is equivalent to the standard form $f(x) = ax^2 + bx + c$. We can convert from the standard form to the vertex form by completing the square. Given $f(x) = ax^2 + bx + c$, we can complete the square to write the function in vertex form.

To convert $f(x) = ax^2 + bx + c$ to vertex form, $f(x) = a(x - h)^2 + k$, we need to complete the square on the quadratic term. We can complete the square to write the function in the form $f(x) = a(x - h)^2 + k$.

Given a quadratic function, the vertex form $f(x) = a(x - h)^2 + k$ provides an easy way to determine the graph’s vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. When the function is written in vertex form, it is easy to identify the vertex because the values of $h$ and $k$ are given.

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry of the parabola is the vertical line $x = h$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$. Therefore, when a quadratic function is given in the form $f(x) = ax^2 + bx + c$, we can find the vertex by using the formula $h = -\frac{b}{2a}$. Then we can find $k$ by evaluating $f(h)$.

The vertex form of a quadratic function $f(x) = a(x - h)^2 + k$ is equivalent to the standard form $f(x) = ax^2 + bx + c$. We can convert from the standard form to the vertex form by completing the square. Given $f(x) = ax^2 + bx + c$, we can complete the square to write the function in vertex form.

To convert $f(x) = ax^2 + bx + c$ to vertex form, $f(x) = a(x - h)^2 + k$, we need to complete the square on the quadratic term. We can complete the square to write the function in the form $f(x) = a(x - h)^2 + k$.

Given a quadratic function, the vertex form $f(x) = a(x - h)^2 + k$ provides an easy way to determine the graph’s vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. When the function is written in vertex form, it is easy to identify the vertex because the values of $h$ and $k$ are given.

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry of the parabola is the vertical line $x = h$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$. Therefore, when a quadratic function is given in the form $f(x) = ax^2 + bx + c$, we can find the vertex by using the formula $h = -\frac{b}{2a}$. Then we can find $k$ by evaluating $f(h)$.

The vertex form of a quadratic function $f(x) = a(x - h)^2 + k$ is equivalent to the standard form $f(x) = ax^2 + bx + c$. We can convert from the standard form to the vertex form by completing the square. Given $f(x) = ax^2 + bx + c$, we can complete the square to write the function in vertex form.

To convert $f(x) = ax^2 + bx + c$ to vertex form, $f(x) = a(x - h)^2 + k$, we need to complete the square on the quadratic term. We can complete the square to write the function in the form $f(x) = a(x - h)^2 + k$.

Given a quadratic function, the vertex form $f(x) = a(x - h)^2 + k$ provides an easy way to determine the graph’s vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. When the function is written in vertex form, it is easy to identify the vertex because the values of $h$ and $k$ are given.

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The axis of symmetry of the parabola is the vertical line $x = h$.

Given the equation $g(x) = 13 + x^2 - 6x$, write the equation in general form and then in standard form. The general form is $g(x) = x^2 - 6x + 13$. The standard form is $g(x) = (x - 3)^2 + 4$. The vertex is $(3, 4)$.

Find the domain and range of $f(x) = -5x^2 + 9x - 1$. The domain is all real numbers. The range is $f(x) \le \frac{61}{20}$, or $(-\infty, \frac{61}{20}]$.

Find the vertex of the quadratic function $f(x) = 2x^2 - 6x + 7$. The vertex is $(-\frac{3}{2}, \frac{13}{2})$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. Using the vertex to determine the shifts, $h = -\frac{3}{2}$ and $k = \frac{13}{2}$.

Write an equation for the quadratic function $g$ in the graph as a transformation of $f(x) = x^2$, and then expand the formula, and simplify terms to write the equation in general form. The equation for the graph is $g(x) = \frac{1}{2}(x + 2)^2 - 3$. Expanding and simplifying, we get $g(x) = \frac{1}{2}x^2 + 2x + 1$.

Given a quadratic function in general form, find the vertex of the parabola. Identify $a$, $b$, and $c$. Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$ into $h = -\frac{b}{2a}$. Find $k$, the $y$-coordinate of the vertex, by evaluating $k = f(h) = f(-\frac{b}{2a})$.

Given a quadratic function, find the domain and range. Identify the domain of any quadratic function as all real numbers. Determine whether $a$ is positive or negative. If $a$ is positive, the parabola has a minimum. If $a$ is negative, the parabola has a maximum. Determine the maximum or minimum value of the parabola, $k$. If the parabola has a minimum, the range is given by $f(x) \ge k$, or $[k, \infty)$. If the parabola has a maximum, the range is given by $f(x) \le k$, or $(-\infty, k]$.

The domain of any quadratic function is all real numbers. The range of a quadratic function written in general form $f(x) = ax^2 + bx + c$ with a positive $a$ value is $f(x) \ge f(-\frac{b}{2a})$, or $[f(-\frac{b}{2a}), \infty)$; the range of a quadratic function written in general form with a negative $a$ value is $f(x) \le f(-\frac{b}{2a})$, or $(-\infty, f(-\frac{b}{2a})]$.

The range of a quadratic function written in standard form $f(x) = a(x - h)^2 + k$ with a positive $a$ value is $f(x) \ge k$; the range of a quadratic function written in standard form with a negative $a$ value is $f(x) \le k$.

The standard form is useful for determining how the graph is transformed from the graph of $y = x^2$. If $k > 0$, the graph shifts upward, whereas if $k < 0$, the graph shifts downward. If $h > 0$, the graph shifts toward the right and if $h < 0$, the graph shifts to the left. The magnitude of $a$ indicates the stretch of the graph. If $|a| > 1$, the point associated with a particular $x$-value shifts farther from the $x$-axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a| < 1$, the point associated with a particular $x$-value shifts closer to the $x$-axis, so the graph appears to become wider, but in fact there is a vertical compression.

The axis of symmetry is defined by $x = -\frac{b}{2a}$. If we use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve $ax^2 + bx + c = 0$ for the $x$-intercepts, or zeros, we find the value of $x$ halfway between them is always $x = -\frac{b}{2a}$, the equation for the axis of symmetry.

The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $(-3, 0)$ and $(-1, 0)$.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. For the linear terms to be equal, the coefficients must be equal. This is the axis of symmetry we defined earlier. Setting the constant terms equal, we can solve for $k$. In practice, though, it is usually easier to remember that $k$ is the output value of the function when the input is $h$, so $f(h) = k$.

Given a quadratic function $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$. The axis of symmetry is $x = h$. The parabola opens upward if $a > 0$ and downward if $a < 0$. The $y$-intercept is $f(0) = a(0 - h)^2 + k = ah^2 + k$. The $x$-intercepts are found by solving $0 = a(x - h)^2 + k$.

The quadratic function $f(x) = a(x - h)^2 + k$ can be written in the form $f(x) = ax^2 + bx + c$ by expanding the squared term and combining like terms. We get $f(x) = a(x^2 - 2hx + h^2) + k = ax^2 -


To find the integral ∫[1,3] 12(1+5x)^5 dx, the usual method is to apply the change-of-variable rule. Let u = 1 + 5x, then du/dx = 5, and du = 5dx. This transforms the integral into a more manageable form, allowing for normal integration.

Alternatively, one can expand the brackets and integrate the resulting function, although this method is more prone to errors and time-consuming. The advantage of this approach is that it can be used to verify the result obtained using the change-of-variable rule. However, in tests and exams, the first method is preferred due to its efficiency and accuracy.


The Golden Ratio is a mathematical constant that can be observed in a rectangular piece of paper when the largest possible square is cut from it, resulting in a rectangle with the same proportions as the original. This ratio has a value of $(1+\sqrt5)/2$, approximately 1.618. The inverse of the Golden Ratio is obtained by subtracting 1, yielding $\phi^{-1} = 1/\phi$. Rearranging this equation gives $\phi^2 - \phi - 1 = 0$, a quadratic equation with solutions $\phi$ and $1/\phi$.

The Golden Ratio can be represented as a continued fraction:
$\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + ...}}}}}$. This infinite representation indicates that the Golden Ratio is an irrational number, as rational numbers have finite continued fractions.

The ratio of successive terms in the Fibonacci sequence approaches the Golden Ratio. The sequence of ratios is:
$\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, \frac{21}{13}, \frac{34}{21}, ...$
These ratios are also obtained by successive truncation of the continued fraction representation of the Golden Ratio.


Let A and B be non-empty subsets of a given set.

a) Given a function f:A→B that is onto, we can construct a function g:B→A that is one-to-one. Since f is onto, for every b in B, there exists an a in A such that f(a) = b. We can choose one such a for each b and define g(b) = a. This ensures g is one-to-one because different elements in B map to different elements in A due to the choice of a unique a for each b.

b) Given a function f:A→B that is one-to-one, constructing a function g:B→A that is onto is not always possible without additional conditions, such as A and B being finite or having the same cardinality. If such conditions are met, for every b in B, if there exists an a in A such that f(a) = b, we define g(b) = a. For elements in B that are not in the range of f, if A and B have the same cardinality or B is finite, we can define g for these elements in a way that covers all of A, ensuring g is onto. However, without these conditions, it's not guaranteed that we can construct g:B→A to be onto using f:A→B that is one-to-one.


To find a Calculus test-taker with expertise in calculus applications in space tourism market analysis, it's essential to understand the role of calculus in real-life applications. Calculus is a branch of mathematics that deals with the study of continuous change, and it has numerous applications in various fields, including space tourism.

In space tourism market analysis, calculus can be used to model and analyze complex systems, such as the trajectory of a spacecraft or the optimization of resources. A Calculus test-taker with expertise in this area should be able to apply calculus concepts, such as differentiation and integration, to solve problems related to space tourism.

To utilize calculus knowledge in space tourism market analysis, one needs to have a strong foundation in calculus concepts, including limits, derivatives, and integrals. Additionally, knowledge of mathematical modeling, data analysis, and computational tools, such as Mathematica or MATLAB, can be beneficial.

When taking an online exam or working on a calculus-related project, it's crucial to understand how to apply calculus concepts to real-world problems. This involves breaking down complex problems into manageable parts, identifying the relevant calculus concepts, and using computational tools to visualize and analyze data.

Some key concepts in calculus that are relevant to space tourism market analysis include:

* Differentiation: used to model rates of change and optimize functions
* Integration: used to model accumulation and calculate quantities such as area and volume
* Mathematical modeling: used to represent complex systems and make predictions
* Data analysis: used to interpret and visualize data related to space tourism

By mastering these concepts and applying them to real-world problems, a Calculus test-taker can demonstrate their expertise in calculus applications in space tourism market analysis.

In terms of resources, there are various books and online materials available that can help individuals develop their calculus skills. Some popular options include calculus textbooks, online courses, and computational software such as Mathematica and MATLAB. These resources can provide a comprehensive introduction to calculus concepts and help individuals develop the skills they need to apply calculus to real-world problems.

Overall, finding a Calculus test-taker with expertise in calculus applications in space tourism market analysis requires a deep understanding of calculus concepts and their applications in real-world problems. By mastering these concepts and applying them to space tourism market analysis, individuals can demonstrate their expertise and provide valuable insights to organizations operating in this field.


**Upcoming Events and Courses**

Next semester, a Seminar on Algorithmic Game Theory will be offered. Interested participants should contact us to agree on a weekly time. Additionally, a lecture on Algorithms and Uncertainty will take place, and interested students are invited to attend.

**Exam Information**

The first round of exams is scheduled for July 31, August 1, August 2, and August 3. Students should contact Matthias to be assigned a time slot.

**Exam Preparation**

Key learning objectives include understanding concepts, definitions, and connections between theorems. Students should be able to explain these concepts and the ideas behind proofs, rather than reproducing tedious calculations. The exam can be taken in English, German, or a combination of both languages. Practicing responses to questions with fellow students is highly recommended, especially for those preparing independently.

**Course Schedule**

| When | Where | Start | Lecturer |
| --- | --- | --- | --- |
| Monday, 10:15-11:45, Friday, 12:15-13:45 | CP1-HSZ / HS 3 | April 9th | Kesselheim |

**Introduction to Algorithmic Game Theory**

In modern computer networks, participants often act strategically, as seen in internet service providers, cloud-based services, and advertising. Algorithmic game theory, a field combining game theory and algorithm design, provides answers to novel questions arising from these interactions. This course introduces students to the foundations of algorithmic game theory, including basic game theory, computability and hardness of equilibria, convergence of dynamics, and designing incentive-compatible auctions.

**Prerequisites and Resources**

Students should have a solid background in algorithms and calculus, but no prior knowledge of game theory is required. Recommended resources include:

* Lecture Notes (updated Jul 13, 2018)
* Algorithmic Game Theory (edited by Nisan, Roughgarden, Tardos, and Vazirani)
* Twenty Lectures on Algorithmic Game Theory (by Tim Roughgarden)
* Game Theory, Alive! (by Karlin and Peres)
* Multiagent Systems (by Shoham and Leyton-Brown)
* Lecture notes on Algorithmic Mechanism Design (by Jason Hartline)


To prove that A\B = (U\B) ∩ (U\A), we can follow these steps:

1. Let x be an element of A\B. This means x belongs to A and x does not belong to B.
   - As x belongs to A, x does not belong to U\A (since U\A contains elements not in A).
   - As x does not belong to B, x belongs to U\B (since U\B contains elements not in B).
   - Therefore, x belongs to (U\B) ∩ (U\A) because it satisfies both conditions of belonging to U\B and not belonging to U\A.

2. Let x be an element of (U\B) ∩ (U\A). This means x belongs to U\B and x does not belong to U\A.
   - As x belongs to U\B, x does not belong to B (by definition of U\B).
   - As x does not belong to U\A, x belongs to A (since U\A contains elements not in A, and not being in U\A means being in A).
   - Therefore, x belongs to A\B because it belongs to A and does not belong to B.

Given these two points, we can conclude that A\B belongs to (U\B) ∩ (U\A) and (U\B) ∩ (U\A) belongs to A\B. Thus, A\B = (U\B) ∩ (U\A).


The NPV formula in Excel can be counterintuitive, especially when it comes to selecting the cash flow. To get the accurate NPV value, Excel requires you to select only the future flows and then discount the initial investment from the result.

In Excel, the NPV formula assumes that the first period used is not the "zero" year, but one year in advance. This means that the cash flow does not include the initial investment. For example, if the cash flow includes an initial investment of "-100,000" in period "zero", this value should not be included in the actual Excel NPV formula. Instead, only the following values should be included, and then the initial investment should be subtracted from the result.

The NPV formula in Excel is called "NPV" and stands for Net Present Value. It calculates the present value of a discounted cash flow, including years with positive and negative flows. The cash flow usually includes the initial investment, but in Excel, this value is not included in the formula.

To use the NPV formula, two key pieces of data are required:

1. The cash flow (excluding the initial investment)
2. The discount rate (usually the rate of return of the best alternative investment with a similar risk profile)

Many Excel users make the mistake of including the initial investment in the actual NPV formula, which can result in a minor difference in the calculated NPV value. However, this difference can be significant, and it is essential to use the correct method to get an accurate NPV value.

To use the NPV formula in Excel, follow these steps:

### Step 1: Insert your Cash Flow

Insert all the cash flows, which can be calculated annually, monthly, or using another time period. Ensure that the discount rate corresponds to the time period used in the cash flow. For example, if the cash flow is calculated annually, use a discount rate expressed in years. To convert a yearly rate to a monthly rate, use the formula:

=(1+A1)^(1/12)-1

Where A1 is the cell containing the yearly rate.

### Step 2: Insert the Discount Rate

Enter the discount rate, which should correspond to the time period used in the cash flow.

### Step 3: Enter the NPV Excel Formula

Insert the NPV formula using the following syntax:

=NPV(E3,B5:B14)+B4

Where:

* E3 is the cell containing the discount rate
* B5:B14 is the range containing the future cash flows
* B4 is the cell containing the initial investment

By following these steps and using the correct method, you can calculate the accurate NPV value in Excel. Remember to exclude the initial investment from the cash flow range and add it to the result separately.


## Intro

Attractors and repellers characterize a system's tendency to gravitate toward or away from a state. These concepts help predict advanced system behavior. For example, big cities act as attractors, gathering people through urban revolution, making it easier to guess future population distribution. A pendulum illustrates this concept, with the bottom position as an attractor and the top as a repeller. Analyzing complex system equations can be challenging, so differentials are used to observe tiny changes in variables and their effects.

## Concept

Consider a function of two variables. Taking a small step along the axes changes the function value by Δf, consisting of two parts: one from Δx and one from Δy. The derivative measures the change per unit traveled. Walking Δx along the x-axis changes the function by ∂f/∂x * Δx, and walking Δy along the y-axis changes it by ∂f/∂y * Δy. The total change is Δf = ∂f/∂x * Δx + ∂f/∂y * Δy.

## Math

The differential of a function is df = ∂f/∂x * dx + ∂f/∂y * dy. This means that taking a small step (dx, dy) changes the function value depending on the partial derivatives.

## Differentials

### Definition of differentials

In multivariable calculus, differentials approximate the change in function value for a function f as a small step is taken in the direction of (dx, dy). The differential, denoted as df, is defined as df = ∂f/∂x * dx + ∂f/∂y * dy. This approximation consists of the partial derivatives multiplied by the corresponding component in the vector. Intuitively, the approximation of the change consists of the rate of change in each direction multiplied by the length of the small step.

### In two variables

Consider a function surface f(x, y) = z. Zooming in on a small square, the differential can be constructed through pictures. The change in function value as the displacement is made in the x-y plane can be approximated by df. The differential of the two-variable function is df = ∂f/∂x * dx + ∂f/∂y * dy. Geometrically, the differential can be interpreted as an approximation of the change in function value by its tangent plane.

## Differentiability

For a single-variable function, differentiability refers to the existence of the derivative. For functions of more variables, the existence of all first partial derivatives is not enough; a linear approximation is required. A function of two variables is differentiable wherever there is a non-vertical tangent plane. The linear approximation of a function at a point is formed by a tangent plane. Moving short distances away from the point, the difference between the function's real value and its linear approximation is negligible. A function is differentiable at a point if the difference between the function's real value and its linear approximation is negligible for short distances. This definition can be extended to functions of more variables, where linear approximations consist of hyperplanes.


# Infinity

**Infinity** is the concept of an object that is larger than any number. It can also describe an object that is smaller than any number when used in the context of "infinitely small." Infinity is not a number, but an abstract concept. Treating infinity as a number without special care can lead to paradoxes.

Infinity is often used to describe the cardinality of a set or object with an infinite number of elements. However, this can lead to nonintuitive results, such as the set of integers and the set of even integers having the same size despite one being contained within the other.

Infinity is denoted by the symbol $\infty$ and is used to describe the limiting behavior of functions, where a function "approaching infinity" means it grows without bound. For example, $f(x) = x^2$ approaches infinity as $x$ grows large, but $f(x)=\frac{2x+1}{x+1}$ does not approach infinity as $x$ grows large.

The concept of infinity is crucial in various contexts, including calculus, set theory, geometry, and inequalities. It is used to analyze the effects of an infinitely small change and is essential in understanding the behavior of functions and sets.

## Intuitive Explanation and Popular Usage

Infinity is not a number, but if it were, it would be the largest number. However, such a largest number does not exist, as $n+1$ would always be larger than $n$. Infinity is a concept that describes quantities that go on forever without end. Common visualizations of infinity include the length of a line, the amount of time it would take to travel along a circle, the number of digits in an irrational number, and the number of points within a boundary.

The term "infinity" is sometimes misused to refer to an unusually large number. For instance, the number of possible chess positions is often referred to as infinite, although it is actually around $10^{120}$.

## Arithmetic with Infinity

Since infinity is not a number, it requires special rules of arithmetic to remain consistent. In measure theory, the real numbers are extended by adding $\infty$ and $-\infty$, which satisfy specific rules:

$\begin{aligned} a + \infty &= \infty + a \\ &= \infty &&(\text{for any } a \text{ besides } -\infty)\\\\ a - \infty &= -\infty + a \\ &= -\infty &&(\text{for any } a \text{ besides } \infty)\\\\ a \cdot \infty &= \infty \cdot a \\ &= \infty &&(\text{for positive } a)\\\\ a \cdot \infty &= \infty \cdot a \\ &= -\infty &&(\text{for negative } a)\\\\ \frac{a}{\infty} &=\frac{a}{-\infty}\\ &=0 &&(\text{for real } a)\\\\ \frac{\infty}{a} &= \infty &&(\text{for positive } a)\\\\ \frac{\infty}{a} &= -\infty. &&(\text{for negative } a) \end{aligned}$

Note that $\frac{1}{0}$ is not $\infty$, and operations involving multiple infinities are not generally well-defined.

## Infinity as a Limit

Infinity is often used to refer to an arbitrarily large number, and arithmetic involving infinity can be performed with the convention that $\infty$ represents a number that is as big as necessary. For instance, $\lim_{n \rightarrow \infty} n^2$ represents the limit as $n$ goes to $\infty$ of $n^2$, which is larger than any other number.

Limits can have finite results, such as $\lim_{n \rightarrow \infty} \frac{1}{n} = 0$. This concept is useful when comparing how "quickly" expressions tend to infinity.

## Infinity in terms of Cardinality

The cardinality of a set is the number of elements it contains. A set with an infinite number of elements has infinite cardinality. For example, the set of positive integers, rational numbers, and real numbers all have infinite cardinality.

Intuitively, it would seem that the set of natural numbers has more elements than the set of even natural numbers, but in fact, both sets have the same cardinality. This can be shown using one-to-one correspondences.

There are different kinds of infinities, and the set of real numbers has a larger cardinality than the set of natural numbers.

## Paradoxes involving Infinity

The concept of infinity can lead to counterintuitive results, such as Zeno's paradoxes:

1. **Achilles and the tortoise**: Achilles can never catch up to the tortoise, as he must first cover half the distance, then half of the remaining distance, and so on.
2. **Dichotomy paradox**: Achilles must complete an infinite number of steps to reach the finish line, so he will never reach the final destination.
3. **Arrow paradox**: The arrow is at a fixed position at any instance in time and cannot be moving, so no motion occurs.

These paradoxes can be resolved using a robust framework of propositional logic and calculus.

Additional paradoxes involving infinity include Russell's paradox and Galileo's paradox, which notes that there are the same number of squares as there are numbers, although most numbers are not squares.

Consider the function $f(x)=\sin(x)$ with domain $\mathbb{R}$. Define the sets:

- $A=\left\{ x\in \mathbb R|\sin(x)=-1 \right\}$
- $B=\left\{ x\in \mathbb R|\sin(x)=0 \right\}$
- $C=\left\{ x\in \mathbb R|\sin(x)=1 \right\}$

Which set has the largest cardinality?

The set $B$ has the largest cardinality, as it contains an infinite number of elements, whereas sets $A$ and $C$ contain a countably infinite number of elements.


**Section 8.1 D2 Percents Math in Our World**

**Learning Objectives**
- Find percent increase or decrease
- Evaluate the validity of claims based on percents

**Problems Involving Percents**
The procedure for finding percent increase or decrease involves:
1. Finding the amount of the increase or the decrease
2. Making a fraction
3. Changing the fraction to a percent

**Example 10: Finding a Percent Change**
A large latte sells at a regular price of $3.50. Today it is on sale for $3.00. To find the percent decrease in the price, follow the procedure:
- Step 1: Find the amount of the decrease ($3.50 - $3.00 = $0.50)
- Step 2: Make a fraction ($0.50 / $3.50)
- Step 3: Change the fraction to a percent

**Example 11: Recognizing Misuse of Percents in Advertising**
A department store advertised a 25% reduction on certain merchandise and an additional 10% discount card for the first 200 people. The advertisement claimed this amounted to a 35% reduction. To evaluate the validity of this claim, calculate the actual percent decrease.

**Classwork**
Complete problems 41, 43, 53, 55, 57 on page 428.

**Assignments and Tests**
- Today’s assignment: Practice 6-8/6-9
- Test date: Wednesday, December 15 (Gold)


# Academic Literature for Multi-Factor Decision Making

## Abstract
The article reviews relevant academic literature on Multi-Criterion Decision Making (MCDM) and discusses its relation to Critical Fallibilism (CF). The author found that MCDM and epistemology literatures mutually ignore each other, and CF's decision-making method is a new approach that MCDM researchers are unaware of.

## Introduction
The author wrote an article on Multi-Factor Decision Making Math, explaining mathematical approaches to decision-making problems and introducing a solution involving converting dimensions to related binary dimensions and then multiplying only binary factors. This article discusses the academic literature for decision-making math and compares existing approaches with CF's new approach.

## MCDM Background
The author was unaware of MCDM literature when writing the initial article. MCDM literature is closer to math, economics, business management, and operations research than to philosophy. However, MCDM issues are relevant to epistemology, and more philosophers should read about and comment on MCDM. The author found that MCDM literature discusses the same basic problem as the initial article, but the majority of people using MCDM have ignored the problems with additively combining factors from different dimensions.

## MCDM Literature Review
The author gathered around 70 pieces of MCDM literature and found that most methods have a lot in common. The author's critical arguments apply widely to existing MCDM methods. Some authors, like Chris Tofallis, have advocated for multiplying factors instead of adding them. The author recommends Tofallis' article "Add or Multiply? A Tutorial on Ranking and Choosing with Multiple Criteria" (2013) for its accessible writing and math.

## Geographic Information Studies
The closest thing to a binary multiplication approach comes from Geographic Information Studies (GIS). GIS involves making maps that convey multiple pieces of useful information at the same time. Combining multiple pieces of information into one final result involves some of the same issues as combining multiple factors into one decision. The author found that GIS articles discuss using breakpoints to convert to binary factors, but the perspective is different from CF's view that multiplying binary factors is optimal.

## Normalization
The author discusses normalization, which means getting data onto the same scale, while weighting means adjusting it based on importance. Normalization procedures are in widespread use but are problematic. The author argues that normalization is unnecessary when multiplying factors and that it distorts data in unintuitive ways. The author concludes that normalization is bad and should be avoided.

## Rank Reversal Example
The author provides an example of rank reversal, where adding or removing an alternative from consideration can change which of two other alternatives has a higher score. This example illustrates the problems with normalization.

## Avoiding Normalization
The author argues that CF doesn't normalize and that using MCDM with multiplication instead of addition also lets you avoid normalizing. The author prefers to weight unnormalized data and then sum it, rather than using normalization methods.

## Large Group Decision Making
The author discusses the use of binary factor multiplication in large organizations and argues that there is no difference in philosophical principles for decision-making alone or with a group of any size. However, there are practical difficulties in groups, especially large groups. The author lacks experience working with or in large organizations and hopes that others will contribute explanations of the problems they face or come up with solutions that apply CF ideas.

## Conclusion
The author concludes that CF's binary factor multiplication approach is an important new idea that MCDM scholars haven't considered or criticized. The author argues that MCDM literature broadly ignores epistemology ideas, and philosophers have largely ignored MCDM. Both MCDM scholars and philosophers could learn more from each other.


The square root of 6 can be written as √6.
7 multiplied by the square root of 3 can be written as 7√3.
6 added to 7 multiplied by the square root of 3 can be written as 7√3 + 6.
Therefore, the square root of 6 * (7 multiplied by the square root of 3 + 6) is written as √6(7√3 + 6).
To solve √6(7√3 + 6), multiply the two terms in the bracket separately with the term outside the bracket.
√6 * 7√3 = 7 * (√6 * √3) = 7 √18.
Since √18 = √9 * √2 = 3 * √2,
7 * √18 = 7 * 3 * √2 = 21 * √2.
Thus, √6 * 7√3 = 21√2.
Also, √6 * 6 = 6√6.
So, √6(7√3 + 6) = (√6 * 7√3) + (√6 * 6) = 21√2 + 6√6.
The roots cannot be simplified further, but you may wish to factorize:
21√2 = 3 * 7√2 and
6√6 = 3 * 2√6.
Then, 21√2 + 6√6 = 3(7√2 + 2√6).


Ohm's law is extremely useful in electronics. The law was found by Georg Ohm and is based on how voltage, current, and resistance are related.

If you increase the voltage in a circuit while the resistance is the same, you get more current. If you increase the resistance in a circuit while the voltage stays the same, you get less current.

Ohm's law is a way of describing the relationship between the voltage, resistance, and current using math: V = RI, where V is the symbol for voltage, I is the symbol for current, and R is the symbol for resistance.

You can switch the formula around and get R = V/I or I = V/R. As long as you have two of the variables, you can calculate the last.

The Ohm's law triangle can be used to remember Ohm's law. To use it, cover the letter you want to find with your hand. If the remaining letters are over each other, it means divide the top one by the bottom one. If they are next to each other, it means multiply one by the other.

For example, to find the formula for voltage, cover the V in the triangle. The R and I are next to each other, so you need to multiply: V = R * I. To find the formula for resistance, cover the R. The V is over the I, so you need to divide: R = V / I. To find the formula for current, cover the I. The V is over the R, so you need to divide: I = V / R.

A simple way to remember Ohm's law is to make an association with it. For example, the "VRIIIIIIII!" rule can be used. Pretend that you are driving a car really fast, then suddenly you hit the brakes really hard. The sound you hear is "VRIIIIIIIIIIII!", which can help you remember V = RI.

A practical example of using Ohm's law is with a circuit that has a battery and a resistor. If the battery is 12 volts and the resistance of the resistor is 600 ohms, you can use the formula I = V/R to calculate the current: I = 12 V / 600 ohm = 0.02 A = 20 mA.

Another example is a circuit with a resistor and a battery, but the voltage of the battery is unknown. If the current in the circuit is 3 mA and the resistance of the resistor is 600 ohms, you can use the formula V = RI to calculate the voltage: V = 600 ohm * 3 mA = 600 ohm * 0.003 A = 1.8 V.

To convert milliamps (mA) to amps (A), divide by 1000. For example, 3 mA is equal to 0.003 A.

Ohm's law can be applied to various electrical components and circuits. It is a fundamental concept in electronics and is used to analyze and design circuits.

In the formula P = VI, power (P) is equal to voltage (V) multiplied by current (I). This formula shows that power is directly proportional to both voltage and current.

Ohm's law states that voltage (V) is directly proportional to current (I), given a constant resistance (R). The formula V = RI shows this relationship.

In summary, Ohm's law is a fundamental concept in electronics that describes the relationship between voltage, current, and resistance. It can be applied to various electrical components and circuits, and is used to analyze and design circuits. The law states that voltage is directly proportional to current, given a constant resistance, and can be expressed mathematically as V = RI.


# Unit Circle Essays

1. Explain what a radian measure represents using the unit circle as a reference.

A radian measure on a unit circle represents the length of the arc at certain points on the unit circle. The unit circle has a circumference of 2π, and an arc of the unit circle has the same length as the measure of the central angle that intercepts that arc.

2. How do special right triangles directly relate to a unit circle?

On the unit circle, the radius of the circle is also the hypotenuse. If the hypotenuse is set to 1, the side (x, y) of the triangle will be the sine and cosine values on the unit circle. This relationship is fundamental to understanding the unit circle and trigonometric functions.

3. Suppose that you did not have a unit circle, but rather a circle of radius 5. Will the angle measures in degrees and/or radians change? Why or why not?

The angle measures in degrees and/or radians would not change. One revolution around any circle is always 2π, so the sine and cosine values of the unit circle would still apply to the circle with a radius of 5. The method of using the unit circle to find trigonometric function values would still be relevant, with the x and y coordinates being scaled by the radius.

4. Suppose that you did not have a unit circle, but rather a circle of radius 5. What do you suppose the x- and y-coordinates will be for that circle in Quadrant I?

The coordinates for Quadrant I would be (5, 0), since the circle has expanded in size but the underlying relationships remain the same. This is analogous to the unit circle, where the coordinates for Quadrant I are (1, 0).

5. Consider the two points in Quadrant I on a circle. What is the special relationship between them?

The special relationship between the two points in Quadrant I is that when a line is drawn through those points, it will form an angle that is in standard position. This angle has a direct relationship with the trigonometric functions, such as sine and cosine.

The unit circle is a circle with its center at the origin and a radius of 1. Its circumference is 2π, and the length of an arc is equal to the measure of the central angle that intercepts that arc. The trigonometric functions sine and cosine have special relevance for the unit circle, where the sine of an angle is the y-coordinate of a point on the circle, and the cosine is the x-coordinate. The unit circle can be drawn according to the equation x^2 + y^2 = 1 or the parametric equations x = cos(s), y = sin(s), where s is the length of the arc starting at (1, 0).


# Harmonics (electrical power)

**Harmonic** voltages and currents in an electric power system result from non-linear electric loads, causing power quality problems. Harmonics lead to increased heating in equipment and conductors, misfiring in variable speed drives, and torque pulsations in motors. Reducing harmonics is desirable.

## Current harmonics

In a normal alternating current power system, the current varies sinusoidally at a specific frequency, usually 50 or 60 hertz. Non-linear loads, such as rectifiers, computers, printers, fluorescent lighting, battery chargers, and variable-speed drives, draw non-sinusoidal currents. These complex waveforms can be deconstructed into simple sinusoids using Fourier series analysis, with frequencies at integer multiples of the fundamental frequency.

### Third-order harmonics

Third-order harmonics are generated in non-linear loads, such as transistors, electrical motors, and non-ideal transformers. In three-phase systems, where each phase is 120 degrees apart, third-order harmonics can add constructively, leading to oscillating currents in the neutral wire. To avoid this, delta connections are used, cycling the current around the connection instead of combining it into the neutral wire.

## Voltage harmonics

Voltage harmonics are mostly caused by current harmonics, which distort the voltage waveform due to source impedance. If the source impedance is small, current harmonics cause only small voltage harmonics.

## Harmonics fundamentals

Harmonics provide a mathematical analysis of distortions to current or voltage waveforms using Fourier series. Harmonics are steady-state distortions that repeat every cycle, differing from transient distortions like spikes, dips, and impulses.

## Total harmonic distortion

Total harmonic distortion (THD) measures the level of harmonic distortion in power systems, defined as the ratio of total harmonics to the fundamental frequency.

## Effects

Harmonics increase the current in the system, particularly the third harmonic, which causes a sharp increase in zero sequence current. This effect requires special consideration in electric system design for non-linear loads. Harmonics also affect electrical equipment, such as motors, which experience increased heating and core losses due to higher frequencies.

### Motors

Electric motors suffer losses due to hysteresis and eddy currents, proportional to the frequency of the current. Harmonics produce higher core losses, leading to increased heating and potentially shortening the motor's life.

### Telephones

In the United States, telephone lines transmit frequencies between 300 and 3400 Hz, which do not interfere with electric power distribution at 60 Hz.

## Sources

Non-linear loads, such as rectifiers, battery chargers, electronic ballasts, variable frequency drives, and switching mode power supplies, cause harmonics. Waveform distortion can be mathematically analyzed to show that it is equivalent to superimposing additional frequency components onto a pure sinewave, which are harmonics of the fundamental frequency.

## References

1. "Harmonics Made Simple". ecmweb.com. Retrieved 2015-11-25.
2. "Power System Harmonics" (PDF). Rockwell Automation. Allen-Bradley.
3. National Electrical Code: "A 3-phase, 4-wire, wye-connected power system used to supply power to nonlinear loads may necessitate that the power system design allow for the possibility of high harmonic neutral currents. (Article 220.61(C), FPN No. 2)"


**Divide and Conquer Strategy and 0/1 Knapsack Problem**

Solve one of the following questions:

1. (a) Write the control abstraction for the Divide and Conquer Strategy and comment on its generalized recurrence equation. (6 marks)
1. (b) Find an optimal solution for the 0/1 Knapsack problem using dynamic programming, given:
   - Number of Objects (n) = 4
   - Knapsack Capacity (M) = 5
   - Weights (W1, W2, W3, W4) = (2, 3, 4, 5)
   - Profits (P1, P2, P3, P4) = (3, 4, 5, 6)
   (6 marks)

**Graph Coloring and Time Complexity**

Solve one of the following questions:

2. (a) Calculate the worst-case time complexity of f(n) = 6n(n^3 - n) + 9n using running time complexity. (6 marks)
2. (b) Write an algorithm for the optimum binary search tree. (6 marks)
2. (c) Explain the Travelling Salesperson Problem using the branch and bound method, with an example. (8 marks)

**Non-Deterministic Algorithms and NP Completeness**

Solve one of the following questions:

3. (a) Write a non-deterministic algorithm for the Clique decision problem. (8 marks)
3. (b) Prove that the Vertex cover problem is NP Complete. (8 marks)

**Randomized Algorithms and Parallel Computing**

Solve one of the following questions:

4. (a) Write a short note on Randomized algorithms. (8 marks)
4. (b) Write a non-deterministic algorithm for sorting elements in non-decreasing order. (8 marks)

**Parallel Algorithms and Graph Problems**

Solve one of the following questions:

5. (a) Explain how graph problems can be solved using parallel algorithms. (8 marks)
5. (b) Write Kruskal's algorithm using parallel computing to find the minimum spanning tree, with an example. (8 marks)

**Parallel Shortest Paths and Odd-Even Merge Sort**

Solve one of the following questions:

6. (a) Write an algorithm for finding Parallel shortest paths and comment on the time complexity of this algorithm. (8 marks)
6. (b) Write an odd-even merge sort algorithm, with an example. (8 marks)

**Dijkstra-Scholten Algorithm and Embedded Systems**

Solve one of the following questions:

7. (a) Give and explain the Dijkstra-Scholten algorithm. (9 marks)
7. (b) What is an Embedded system? Explain Embedded system Scheduling. (9 marks)

**Internet of Things and Algorithms in Software Engineering**

Solve one of the following questions:

8. (a) Define the Internet of Things (IoT) and explain the elements of IoT. (9 marks)
8. (b) Give and explain Algorithms in Software Engineering, with an example. (9 marks)


# Divide by 2 Digit Numbers - 5th Grade Math

In the fifth grade, students divide 4-digit numbers by 2-digit numbers. The **Divide by 2 Digit Numbers Worksheet** focuses on divisions without remainders, where students consider the nearest multiple of the divisor. With 1-digit divisors, this is straightforward due to memorized multiplication facts. However, with 2-digit divisors, estimation strategies are essential for guessing the nearest multiple. The worksheet starts with quotient estimation problems.

For example, when dividing 7290 by 81, students can estimate the quotient by simplifying it to 7200 ÷ 80, which equals 90. This estimation serves as a starting point for the division process.

### Common Core Alignment

**5.NBT.6** requires finding whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors. Students use strategies based on place value, operation properties, and the relationship between multiplication and division. They illustrate and explain calculations using equations, rectangular arrays, and/or area models.


The creator of sites like The Problem Site and Articles for Educators answers questions on various topics.

A pastor inquired about the significance of the number 153, which appears in the Gospel of John, chapter 21. The number 153 has several unique properties: it is a Triangular Number, being the sum of integers from 1 to 17, and a Happy Cube, where the sum of the cubes of its digits equals the number itself (1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153).

This property can be demonstrated with other numbers, such as 24, where applying the Happy Cube process yields a different result (2^3 + 4^3 = 8 + 64 = 72, and further processing leads to 351, not 24). However, 153 is a special case where the Happy Cube process results in the original number.

The significance of 153 in the Gospel of John, chapter 21, is still unclear. Biblical numerologists offer various explanations, and it's likely that 153 numerologists would have 153 different answers. Two possible interpretations are:

1. The number 153 represents the work Jesus did, as some sources claim that, excluding mass miracles, the Gospels record 153 individuals Jesus ministered to. However, this count can be subjective and influenced by how miracles are defined and counted.
2. The number 153 can be seen as 9 x 17, where 9 allegedly represents the Holy Spirit and 17 represents the uttermost parts of the earth. This interpretation suggests that 153 symbolizes going to the uttermost parts of the earth by the power of the Holy Spirit.

These explanations are speculative, and the true significance of 153 in the Gospel of John remains unclear.