Datasets:
jacobcd52
/
hs_math_cleaned

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
235
3.08k
Identifying Polynomial Functions Which of the following are polynomial functions? f (x) = 2x 3 β‹… 3x + 4 g(x) = βˆ’x(x2 βˆ’ 4) h(x) = 5 √ β€” x + 2 Solution The first two functions are examples of polynomial functions because they can be written in the form f (x) = an xn +... + a2 x2 + a1 x + a0, where the powers are non-neg...
4x 3 g(t) = 5t 5 βˆ’ 2t 3 + 7t h(p) = 6p βˆ’ p 3 βˆ’ 2 Solution For the function f (x), the highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, βˆ’4x3. The leading coefficient is the coefficient of that term, βˆ’4. For the function g(t), the highest power of t is 5, so the degree is...
4 5 x f (x) = βˆ’6x3 + 7x2 + 3x + 1 βˆ’6x3 x Table 3 SECTION 3.3 power Functions and polynomial Functions 231 Example 6 Identifying End Behavior and Degree of a Polynomial Function Describe the end behavior and determine a possible degree of the polynomial function in Figure 7. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 –5 –4 –3 –2...
3), so the end behavior is as x β†’ βˆ’βˆž, f (x) β†’ βˆ’βˆž as x β†’ ∞, f (x) β†’ βˆ’βˆž 232 CHAPTER 3 polynomial and rational Functions Try It #5 Given the function f (x) = 0.2(x βˆ’ 2)(x + 1)(x βˆ’ 5), express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function. Identifying ...
substitute 0 for x. The y-intercept is (0, 8). f (0) = (0 βˆ’ 2)(0 + 1)(0 βˆ’ 4) = (βˆ’2)(1)(βˆ’ 4) = 8 SECTION 3.3 power Functions and polynomial Functions 233 The x-intercepts occur when the output is zero. 0 = (x βˆ’ 2)(x + 1)(x βˆ’ 4) The x-intercepts are (2, 0), (–1, 0), and (4, 0). x = 2 or x βˆ’ 2 = 0 or x + 1 = 0 x = βˆ’1 or ...
and (3, 0) x 21 3 4 5 y-intercept (0, βˆ’45) Try It #6 Given the polynomial function f (x) = 2x3 βˆ’ 6x2 βˆ’ 20x, determine the y- and x-intercepts. Figure 11 234 CHAPTER 3 polynomial and rational Functions Comparing Smooth and Continuous Graphs The degree of a polynomial function helps us to determine the number of x-inter...
tells us this is the graph of an even-degree polynomial. See Figure 13. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 x-intercepts 21 3 4 5 x Turning points Figure 13 The graph has 2 x-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be rea...
college.org/l/leastposdegree) 236 CHAPTER 3 polynomial and rational Functions 3.3 SeCTIOn exeRCISeS VeRBAl 1. Explain the difference between the coefficient 2. If a polynomial function is in factored form, what of a power function and its degree. 3. In general, explain the end behavior of a power function with odd degr...
26. g(n) = βˆ’2(3n βˆ’ 1)(2n + 1) 27. f (x) = x4 βˆ’ 16 28. f (x) = x3 + 27 29. f (x) = x(x2 βˆ’ 2x βˆ’ 8) 30. f (x) = (x + 3)(4x2 βˆ’ 1) GRAPHICAl For the following exercises, determine the least possible degree of the polynomial function shown. 31. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 32. 21 3 4 5 x –5 –4 –3 –2 y 5 4 3 2 1...
x4 βˆ’ 5x2 50. f (x) = x5 __ 10 βˆ’ x4 48. f (x) = x2(1 βˆ’ x)2 TeCHnOlOGY For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. 51. f (x) = x3(x βˆ’ 2) 52. f (x) = x(x βˆ’ 3)(x + 3) 53. f (x) = x(14 βˆ’ 2x)(10 βˆ’ 2x) 54. f (x) = x(14 βˆ’ 2x...
β†’ ∞, as x β†’ ∞, f (x) β†’ βˆ’βˆž. 65. The y-intercept is (0, 1). There is no x-intercept. Degree is 4. End behavior: as x β†’ βˆ’βˆž, f (x) β†’ ∞, as x β†’ ∞, f (x) β†’ ∞. ReAl-WORlD APPlICATIOnS For the following exercises, use the written statements to construct a polynomial function that represents the required information. 66. An oi...
48.6 48.7 47.1 The revenue can be modeled by the polynomial function Table 1 R(t) = βˆ’0.037t4 + 1.414t3 βˆ’ 19.777t2 + 118.696t βˆ’ 205.332 where R represents the revenue in millions of dollars and t represents the year, with t = 6 corresponding to 2006. Over which intervals is the revenue for the company increasing? Over ...
. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section: 1. ...
Find the x-intercepts of f (x) = x 3 βˆ’ 5x 2 βˆ’ x + 5. Solution Find solutions for f (x) = 0 by factoring. x 3 βˆ’ 5x 2 βˆ’ x + 5 = 0 Factor by grouping. x 2(x βˆ’ 5) βˆ’ (x βˆ’ 5) = 0 Factor out the common factor. (x 2 βˆ’ 1)(x βˆ’ 5) = 0 Factor the difference of squares. (x + 1)(x βˆ’ 1)(x βˆ’ 5) = 0 Set each factor equal to zero. x + ...
x-intercepts of h(x) = x3 + 4x2 + x βˆ’ 6. Solution This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In thes...
intercepts is different. f (x) = (x + 3)(x βˆ’ 2)2(x + 1)3 2 4 x y 40 30 20 10 –10 –20 –30 –40 –4 –2 Figure 7 Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero. The x-intercept x = βˆ’3 is the solution of equation (x + 3) = 0. The graph passes directly through the x-intercept...
powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the x-axis. For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each incre...
icity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6. Try It #2 Use the graph of the function of degree 5 in Figure 10 to identify the zeros of the function and their multiplicities. y 60 40 20 –6 –4 –2 2 4 6 x –20 –40 –60 Figure 10 Determining end ...
polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function f (x) = x4 βˆ’ x3 βˆ’ 4x2 + 4x in Figure 12. The graph has three turning points. y Increa...
. How To… Given a polynomial function, sketch the graph. 1. Find the intercepts. 2. Check for symmetry. If the function is an even function, its graph is symmetrical about the y-axis, that is, f (βˆ’x) = f (x). If a function is an odd function, its graph is symmetrical about the origin, that is, f (βˆ’x) = βˆ’f (x). 3. Use t...
-axis, so the function must start increasing. At (0, 90), the graph crosses the y-axis at the y-intercept. See Figure 14. y (0, 90) (βˆ’3, 0) x Figure 14 Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0). ...
, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis. Figure 17 shows that there is a zero between a and b. y 5 4 3 2 1 –1–1 –2 –3 –4 –5 –5 –4 –3 –2 f (b) is positive f (c) = 0 x 5 21 3 4 f(a) is negative Figure 1...
each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors. factored form of polynomials If a polynomial of lowest degree p has horizontal intercepts at x = x1, x2, …, xn, then the polynomial can be written (x βˆ’ x2)p2 … (x βˆ’ xn)pn whe...
0 + 3)(0 βˆ’ 2)2(0 βˆ’ 5) βˆ’2 = βˆ’60a The graphed polynomial appears to represent the function f (x) = (x + 3)(x βˆ’ 2)2 (x βˆ’ 5). a = 1 __ 30 1 __ 30 Try It #5 Given the graph shown in Figure 20, write a formula for the function shown. y 12 8 4 –6 –4 –2 2 4 6 x –4 –8 –12 Figure 20 Using Local and Global Extrema With quadratics...
a global minimum or maximum? No. Only polynomial functions of even degree have a global minimum or maximum. For example, f (x) = x has neither a global maximum nor a global minimum. Example 11 Using Local Extrema to Solve Applications An open-top box is to be constructed by cutting out squares from each corner of a 14...
24. V(w) 340 339 338 337 336 335 334 333 332 331 330 2.4 2.6 2.8 3 Figure 24 w From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side. Try It #6 Use technology to find the maximum and minimum values on the interval [...
βˆ’ 4 16. f (x) = x3 + 2x2 βˆ’ 9x βˆ’ 18 17. f (x) = 2x3 βˆ’ x2 βˆ’ 8x + 4 18. f (x) = x6 βˆ’ 7x3 βˆ’ 8 19. f (x) = 2x4 + 6x2 βˆ’ 8 20. f (x) = x3 βˆ’ 3x2 βˆ’ x + 3 21. f (x) = x6 βˆ’ 2x4 βˆ’ 3x2 22. f (x) = x6 βˆ’ 3x4 βˆ’ 4x2 23. f (x) = x5 βˆ’ 5x3 + 4x For the following exercises, use the Intermediate Value Theorem to confirm that the given poly...
f (x) = 2x4(x3 βˆ’ 4x2 + 4x) 39. f (x) = 4x5 βˆ’ 12x4 + 9x3 41. f (x) = 4x4(9x4 βˆ’ 12x3 + 4x2) GRAPHICAl For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior. 42. f (x) = (x + 3)2(x βˆ’ 2) 43. g(x) = (x + 4)(x βˆ’ 1)2 44. h(x) = (x βˆ’ 1)3(x + 3)2 45. k(x) = (x βˆ’ 3...
4 3 2 1 –1–1 –2 –3 –4 –5 y 5 4 3 2 1 –1–1 –2 –3 –4 –5 –5 –4 –3 –2 –5 –4 –3 –2 For the following exercises, use the given information about the polynomial graph to write the equation. 57. Degree 3. Zeros at x = βˆ’2, x = 1, and x = 3. 58. Degree 3. Zeros at x = βˆ’5, x = βˆ’2, and x = 1. y-intercept at (0, βˆ’4). y-intercept a...
x4 + 3x βˆ’ 2 71. f (x) = x4 βˆ’ x3 + 1 exTenSIOnS For the following exercises, use the graphs to write a polynomial function of least degree. (0, 50,000,000) 100 200 300 400 500 600 700 x 6Β·107 5Β·107 4Β·107 3Β·107 2Β·107 1Β·107 –100 0 –1Β·107 –2Β·107 –3Β·107 –4Β·107 –5Β·107 –6Β·107 –7Β·107 72. f (x) 24 16 73. f (x), 0 2 3 (0, 8) 4 3...
CTIVeS In this section, you will: β€’ Use long division to divide polynomials. β€’ Use synthetic division to divide polynomials. 3.5 DIVIDInG POlYnOMIAlS Figure 1 lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr) The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with len...
lynomial and rational Functions Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic. dividend = (divisor β‹… quotient) + remainder 178 = (3 β‹… 59) + 1 = 177 + 1 = 178 We call this the Division Algorithm and will di...
18 βˆ’ 31 _ x + 2 2x3 βˆ’ 3x2 + 4x + 5 __ x + 2 = (x + 2)(2x2 βˆ’ 7x + 18) βˆ’ 31 We can identify the dividend, the divisor, the quotient, and the remainder. 2x3 – 3x2 + 4x + 5 = (x + 2) (2x2 – 7x + 18) + (–31) Dividend Divisor Quotient Remainder Writing the result in this manner illustrates the Division Algorithm. SECTION 3....
βˆ’ 2 5x2 divided by x is 5x. 5x x + 1)5x2 + 3x βˆ’ 2 βˆ’(5x2 + 5x) βˆ’2x βˆ’ 2 5x βˆ’ 2 x + 1)5x2 + 3x βˆ’ 2 βˆ’(5x2 + 5x) βˆ’2x βˆ’ 2 βˆ’(βˆ’2x βˆ’ 2) 0 Multiply x + 1 by 5x. Subtract. Bring down the next term. βˆ’2x divided by x is βˆ’2. Multiply x + 1 by βˆ’2. Subtract. The quotient is 5x βˆ’ 2. The remainder is 0. We write the result as or 5x2 + ...
+ 1 = 6x3 + 11x2 βˆ’ 31x + 15 Notice, as we write our result, β€’ the dividend is 6x3 + 11x2 βˆ’ 31x + 15 β€’ the divisor is 3x βˆ’ 2 β€’ the quotient is 2x2 + 5x βˆ’ 7 β€’ the remainder is 1 Try It #1 Divide 16x3 βˆ’ 12x2 + 20x βˆ’ 3 by 4x + 5. Using Synthetic Division to Divide Polynomials As we’ve seen, long division of polynomials ca...
7x + 18 and the remainder is βˆ’31. The process will be made more clear in Example 3. synthetic division Synthetic division is a shortcut that can be used when the divisor is a binomial in the form x βˆ’ k. In synthetic division, only the coefficients are used in the division process. How To… Given two polynomials, use sy...
20 10 βˆ’8 βˆ’4 20 2 βˆ’10 0 The result is 4x 2 + 2x βˆ’ 10. The remainder is 0. Thus, x + 2 is a factor of 4x3 + 10x2 βˆ’ 6x βˆ’ 20. Analysis The graph of the polynomial function f (x) = 4x3 + 10x2 βˆ’ 6x βˆ’ 20 in Figure 2 shows a zero at x = k = βˆ’2. This confirms that x + 2 is a factor of 4x 3 + 10x2 βˆ’ 6x βˆ’ 20. βˆ’2 βˆ’1.8 –5 –4 –3 –2 ...
x4 βˆ’ 3x3 βˆ’ 33x2 + 54x = 3x β‹… (x βˆ’ 2) β‹… h To solve for h, first divide both sides by 3x. 3x β‹… (x βˆ’ 2) β‹… h ____________ 3x = 3x4 βˆ’ 3x3 βˆ’ 33x2 + 54x ___________________ 3x Now solve for h using synthetic division. (x βˆ’ 2)h = x3 βˆ’ x2 βˆ’ 11x + 18 h = x3 βˆ’ x2 βˆ’ 11x + 18 ________________ x βˆ’ 2 2 1 βˆ’1 βˆ’11 2 1 18 2 βˆ’18 βˆ’9 0 1 Th...
+ 2) 9. (2x2 βˆ’ 3x + 2) Γ· (x + 2) 12. (x3 βˆ’ 3x2 + 5x βˆ’ 6) Γ· (x βˆ’ 2) 4. (2x2 βˆ’ 9x βˆ’ 5) Γ· (x βˆ’ 5) 7. (6x2 βˆ’ 25x βˆ’ 25) Γ· (6x + 5) 10. (x3 βˆ’ 126) Γ· (x βˆ’ 5) 13. (2x3 + 3x2 βˆ’ 4x + 15) Γ· (x + 3) 5. (3x2 + 23x + 14) Γ· (x + 7) 8. (βˆ’x2 βˆ’ 1) Γ· (x + 1) 11. (3x2 βˆ’ 5x + 4) Γ· (3x + 1) For the following exercises, use synthetic divisi...
x3 βˆ’ 2x2 + x βˆ’ 4) Γ· (x + 3) 21. (2x3 + 7x2 βˆ’ 13x βˆ’ 3) Γ· (2x βˆ’ 3) 23. (4x3 βˆ’ 5x2 + 13) Γ· (x + 4) 25. (x3 βˆ’ 21x2 + 147x βˆ’ 343) Γ· (x βˆ’ 7) 27. (9x3 βˆ’ x + 2) Γ· (3x βˆ’ 1) 29. (x4 + x3 βˆ’ 3x2 βˆ’ 2x + 1) Γ· (x + 1) 31. (x4 + 2x3 βˆ’ 3x2 + 2x + 6) Γ· (x + 3) 33. (x4 βˆ’ 8x3 + 24x2 βˆ’ 32x + 16) Γ· (x βˆ’ 2) 35. (x4 βˆ’ 12x3 + 54x2 βˆ’ 108x + 81)...
exercises 265 47. Factor is x2 + x + 1 48. Factor is x2 + 2x + 2 y 60 40 20 y 18 12 6 –6 –4 –2 2 4 6 x –6 –4 –2 2 4 6 x –20 –40 –60 – 6 –12 –18 For the following exercises, use synthetic division to find the quotient and remainder. 49. 4x3 βˆ’ 33 _______ x βˆ’ 2 51. 3x3 + 2x βˆ’ 5 __________ x βˆ’ 1 50. 2x3 + 25 _______ x + 3...
the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. 67. Volume is 12x3 + 20x2 βˆ’ 21x βˆ’ 36, length is 2x + 3, width is 3x βˆ’ 4. 68. Volume is 18x3 βˆ’ 21x2 βˆ’ 40x + 48, length is 3x βˆ’ 4, width is 3x βˆ’ 4. 69. Volume is 10x3 + 27x2 + 2x βˆ’ 24, length i...
will discuss a variety of tools for writing polynomial functions and solving polynomial equations. evaluating a Polynomial Using the Remainder Theorem In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is ...
32 25 6 The remainder is 25. Therefore, f (2) = 25. Analysis We can check our answer by evaluating f (2). f (x) = 6x4 βˆ’ x3 βˆ’ 15x2 + 2x βˆ’ 7 f (2) = 6(2)4 βˆ’ (2)3 βˆ’ 15(2)2 + 2(2) βˆ’ 7 = 25 Try It #1 Use the Remainder Theorem to evaluate f (x) = 2x5 βˆ’ 3x4 βˆ’ 9x3 + 8x2 + 2 at x = βˆ’3. Using the Factor Theorem to Solve a Polyn...
the product of (x βˆ’ k) and the quadratic quotient. 4. If possible, factor the quadratic. 5. Write the polynomial as the product of factors. Example 2 Using the Factor Theorem to Solve a Polynomial Equation Show that (x + 2) is a factor of x3 βˆ’ 6x2 βˆ’ x + 30. Find the remaining factors. Use the factors to determine the ...
(5x βˆ’ 2)(4x βˆ’ 3) Create the quadratic function, multiplying the factors. f (x) = 20x2 βˆ’ 23x + 6 Expand the polynomial. f (x) = (5 β‹… 4)x2 βˆ’ 23x + (2 β‹… 3) Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leadi...
of 2 are q = Β±1, Β±2. If any of the four real zeros are rational zeros, then they will be of one of the following factors of βˆ’4 divided by one of the factors of 2. p 1 1 __ __ __ __ __ __ __ __ = 2, which have already been listed. So we can shorten our list. = 1 and Note that 2 2 p _ q = Factors of the last __ Factors ...
the function. 2. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate. 3. Repeat step two using the quotient found with synthetic division. If possible, c...
x-axis, indicating the odd multiplicity (1, 3, 5…) for the zero x = 1. y Bounce 1.5 1 0.5 –2.5 –2 –1.5 0.5 1 1.5 2 2.5 x Cross –1 –0.5 –0.5 –1 –1.5 –2 –2.5 Figure 1 SECTION 3.6 Zeros oF polynomial Functions 271 Using the Fundamental Theorem of Algebra Now that we can find rational zeros for a polynomial function, we w...
… Does every polynomial have at least one imaginary zero? No. A complex number is not necessarily imaginary. Real numbers are also complex numbers. Example 6 Finding the Zeros of a Polynomial Function with Complex Zeros Find the zeros of f (x) = 3x3 + 9x2 + x + 3. p _ Solution The Rational Zero Theorem tells us that if...
12 6 Cross –6 –4 –2 2 4 6 x –6 –12 –18 Figure 2 Try It #4 Find the zeros of f (x) = 2x3 + 5x2 βˆ’ 11x + 4. Using the linear Factorization Theorem to Find Polynomials with Given Zeros A vital implication of the Fundamental Theorem of Algebra, as we stated above, is that a polynomial function of degree n will have n zeros...
the linear factors to expand the polynomial. 3. Substitute (c, f (c)) into the function to determine the leading coefficient. 4. Simplify. SECTION 3.6 Zeros oF polynomial Functions 273 Example 7 Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros Find a fourth degree polynomial with real coeff...
Try It #5 Find a third degree polynomial with real coefficients that has zeros of 5 and βˆ’2i such that f (1) = 10. Using Descartes’ Rule of Signs There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending o...
2 βˆ’ 4(βˆ’x) βˆ’ 12 f (βˆ’x) = βˆ’x 4 + 3x 3 + 6x 2 + 4x βˆ’ 12 f (βˆ’x) = βˆ’x 4 + 3x 3 + 6x 2 + 4x βˆ’ 12 Figure 4 Again, there are two sign changes, so there are either 2 or 0 negative real roots. There are four possibilities, as we can see in Table 1. Positive Real Zeros 2 2 0 0 Negative Real Zeros 2 0 2 0 Table 1 Complex Zeros 0 2...
the width, so we can express the height of the cake as h = 1 __ w. Let’s write the volume of the cake in terms of width of the cake. 3 1 __ w ξ€ͺ V = (w + 4)(w) ξ€’ 3 1 4 __ __ V = w 3 + w 2 3 3 Substitute the given volume into this equation. 1 4 __ __ 351 = w 3 + w 2 3 3 1053 = w 3 + 4w 2 Substitute 351 for V. Multiply b...
) β€’ Complex Factorization Theorem (http://openstaxcollege.org/l/factortheorem) β€’ Find the Zeros of a Polynomial Function (http://openstaxcollege.org/l/findthezeros) β€’ Find the Zeros of a Polynomial Function 2 (http://openstaxcollege.org/l/findthezeros2) β€’ Find the Zeros of a Polynomial Function 3 (http://openstaxcolleg...
x3 + x2 βˆ’ 20x + 12; x + 3 17. f (x) = 2x3 + 3x2 + x + 6; x + 2 18. f (x) = βˆ’5x3 + 16x2 βˆ’ 9; x βˆ’ 3 20. 4x3 βˆ’ 7x + 3; x βˆ’ 1 19. x3 + 3x2 + 4x + 12; x + 3 21. 2x3 + 5x2 βˆ’ 12x βˆ’ 30, 2x + 5 For the following exercises, use the Rational Zero Theorem to find all real zeros. 22. x3 βˆ’ 3x2 βˆ’ 10x + 24 = 0 23. 2x3 + 7x2 βˆ’ 10x βˆ’ 24...
75 = 0 45. 2x3 βˆ’ 3x2 + 32x + 17 = 0 GRAPHICAl For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. 46. f (x) = x3 βˆ’ 1 47. f (x) = x4 βˆ’ x2 βˆ’ 1 48. f (x) = x3 βˆ’ 2x2 βˆ’ 5x + 6 49. ...
, construct a polynomial function of least degree possible using the given information. 66. Real roots: βˆ’1, 1, 3 and (2, f (2)) = (2, 4) 67. Real roots: βˆ’1, 1 (with multiplicity 2 and 1) and 1 __ 68. Real roots: βˆ’2, (with multiplicity 2) and 2 (βˆ’3, f (βˆ’3)) = (βˆ’3, 5) 70. Real roots: βˆ’4, βˆ’1, 1, 4 and (βˆ’2, f (βˆ’2)) = (βˆ’2, ...
ptotes. Identify horizontal asymptotes. β€’ Graph rational functions. 3.7 RATIOnAl FUnCTIOnS Suppose we know that the cost of making a product is dependent on the number of items, x, produced. This is given by the equation C(x) = 15,000x βˆ’ 0.1x2 + 1000. If we want to know the average cost for producing x items, we would ...
οΏ½ f (x) β†’ ∞ f (x) β†’ βˆ’βˆž f (x) β†’ a Meaning x approaches a from the left (x < a but close to a) x approaches a from the right (x > a but close to a) x approaches infinity (x increases without bound) x approaches negative infinity (x decreases without bound) The output approaches infinity (the output increases without boun...
4 –3 –2 321 4 5 x x = 0 Figure 3 vertical asymptote A vertical asymptote of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a. We write As x β†’ a, f (x) β†’ ∞, or as x β†’ a, f (x) β†’ βˆ’βˆž. 1 _ x End Behavior of f (x ) = As the values of x approach infinity, th...
= 2. As x β†’ 2βˆ’, f (x) β†’ βˆ’βˆž, and as x β†’ 2+, f (x) β†’ ∞. And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at y = 4. As the inputs increase without bound, the graph levels off at 4. As x β†’ ∞, f (x) β†’ 4 and as x β†’ βˆ’βˆž, f (x) β†’ 4. Try It #...
of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions. rational function A rational function is a function that can be written as the quotient of two polynomial functions P(x) and Q(x)...
the horizontal asymptote is y = 0.1. This means the concentration, C, the ratio of pounds of sugar to gallons of water, will approach 0.1 in the long term. Try It #3 There are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomo...
its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location. Vertical Asymptotes The vertical asymptotes of a rational functi...
βˆ’ 1 __________ x2 βˆ’ 2x βˆ’ 3 f (x) = (x + 1)(x βˆ’ 1) ____________ (x + 1)(x βˆ’ 3) Notice that x + 1 is a common factor to the numerator and the denominator. The zero of this factor, x = βˆ’1, is the location of the removable discontinuity. Notice also that x βˆ’ 3 is not a factor in both the numerator and denominator. The zer...
ote at x = βˆ’2, but at x = 2 the graph will have a hole. Try It #5 Find the vertical asymptotes and removable discontinuities of the graph of f (x) = x2 βˆ’ 25 ___________ x3 βˆ’ 6x2 + 5x. Identifying Horizontal Asymptotes of Rational Functions While vertical asymptotes describe the behavior of a graph as the output gets ve...
level off, so this graph has no horizontal asymptote. However, the graph of g(x) = 3x looks like a diagonal line, and since f will behave similarly to g, it will approach a line close to y = 3x. This line is a slant asymptote. SECTION 3.7 rational Functions 287 To find the equation of the slant asymptote, divide slant...
, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote. It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of ...
x2 βˆ’ 4x + 1 _ x + 2 at x2 βˆ’ 4x + 1 _ x + 2. 2 1 βˆ’4 βˆ’2 1 βˆ’6 1 12 13 The quotient is x βˆ’ 2 and the remainder is 13. There is a slant asymptote at y = x βˆ’ 2. c. k(x) = : The degree of p = 2 < degree of q = 3, so there is a horizontal asymptote y = 0. x2 + 4x _ x3 βˆ’ 8 Example 8 Identifying Horizontal Asymptotes In the sug...
tend towards zero as the inputs get large, and so as x β†’ ±∞, f (x) β†’ 0. This function will have a horizontal asymptote at y = 0. See Figure 15. y 6 4 2 y = 0 –6 –4 –2 2 4 6 8 x –2 –4 –6 x = βˆ’2 x = 1 x = 5 Figure 15 SECTION 3.7 rational Functions 289 Try It #6 Find the vertical and horizontal asymptotes of the function...
3 –4 –5 –6 x = βˆ’2 Figure 16 Try It #7 Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x- and y-intercepts and the horizontal and vertical asymptotes. 290 CHAPTER 3 polynomial and rational Functions Graphing Rational Functions In Exa...
factor. β€’ At the x-intercept x = 3 corresponding to the (x βˆ’ 3) factor of the numerator, the graph passes through the axis as we would expect from a linear factor. β€’ At the vertical asymptote x = βˆ’3 corresponding to the (x + 3)2 factor of the denominator, the graph heads towards positive infinity on both sides of the ...
. Setting each factor equal to zero, we find x-intercepts at x = βˆ’2 and x = 3. At each, the behavior will be linear (multiplicity 1), with the graph passing through the intercept. We have a y-intercept at (0, 3) and x-intercepts at (βˆ’2, 0) and (3, 0). To find the vertical asymptotes, we determine when the denominator i...
), use the characteristics of polynomials and rational functions to describe its behavior and sketch the function. Writing Rational Functions Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A ra...
powers. 3. Use any clear point on the graph to find the stretch factor. Example 12 Writing a Rational Function from Intercepts and Asymptotes Write an equation for the rational function shown in Figure 22. –6 –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 –7 321 4 5 6 x Figure 22 Solution The graph appears to have x-int...
interasymptote) SECTION 3.7 section exercises 295 3.7 SeCTIOn exeRCISeS VeRBAl 1. What is the fundamental difference in the algebraic representation of a polynomial function and a rational function? 3. If the graph of a rational function has a removable discontinuity, what must be true of the functional rule? 5. Can a ...
βˆ’ 10x + 24 24. f (x) = 94 βˆ’ 2x2 _______ 3x2 βˆ’ 12 For the following exercises, describe the local and end behavior of the functions. 25. f (x) = x _____ 2x + 1 26. f (x) = 2x _____ x βˆ’ 6 27. f (x) = βˆ’2x _____ x βˆ’ 6 28. f (x) = x2 βˆ’ 4x + 3 _________ x2 βˆ’ 4x βˆ’ 5 29. f (x) = 2x2 βˆ’ 32 ___________ 6x2 + 13x βˆ’ 5 For the foll...
x) = 2x 2 + x βˆ’ 1 _ x βˆ’ 4 48. k(x) = 2x 2 βˆ’ 3x βˆ’ 20 __ x βˆ’ 5 49. w(x) = (x βˆ’ 1)(x + 3)(x βˆ’ 5) __ (x + 2)2(x βˆ’ 4) 50. z(x) = (x + 2)2(x βˆ’ 5) __ (x βˆ’ 3)(x + 1)(x + 4) For the following exercises, write an equation for a rational function with the given characteristics. 51. Vertical asymptotes at x = 5 and x = βˆ’5, x-inter...
642 8 10 x 61. 642 8 10 x –10 –8 –6 –4 64. 642 8 10 x –10 –8 –6 –4 y 5 4 3 2 1 –2 –1 –2 –3 –4 –5 y 5 4 3 2 1 –2 –1 –2 –3 –4 –5 60. 63. y 5 4 3 2 1 –2 –1 –2 –3 –4 –5 y 5 4 3 2 1 –2 –1 –2 –3 –4 –5 –10 –8 –6 –4 –10 –8 –6 –4 nUMeRIC For the following exercises, make tables to show the behavior of the function near the ver...
contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes. 81. A lar...
. Let x = radius. SECTION 3.8 inverses and radical Functions 299 leARnInG OBjeCTIVeS In this section, you will: β€’ β€’ Find the inverse of a polynomial function. Restrict the domain to find the inverse of a polynomial function. 3.8 InVeRSeS AnD RADICAl FUnCTIOnS A mound of gravel is in the shape of a cone with the height ...
2 –2 –4 –6 –10 –8 –6 –4 642 8 10 x Figure 3 From this we find an equation for the parabolic shape. We placed the origin at the vertex of the parabola, so we know the equation will have form y(x) = ax2. Our equation will need to pass through the point (6, 18), from which we can solve for the stretch factor a. Our parabo...
omial functions, some basic polynomials do have inverses. Such functions are called invertible functions, and we use the notation f βˆ’1(x). Warning: f βˆ’1(x) is not the same as the reciprocal of the function f (x). This use of β€œβˆ’1” is reserved to denote inverse functions. To denote the reciprocal of a function f (x), we ...
inverses. x Try It #1 Show that f (x) = x + 5 _____ 3 and f βˆ’1(x) = 3x βˆ’ 5 are inverses. 30 2 CHAPTER 3 polynomial and rational Functions Example 2 Finding the Inverse of a Cubic Function Find the inverse of the function f (x) = 5x3 + 1. Solution This is a transformation of the basic cubic toolkit function, and based ...
order to find their inverses. restricting the domain If a function is not one-to-one, it cannot have an inverse. If we restrict the domain of the function so that it becomes one-to-one, thus creating a new function, this new function will have an inverse. SECTION 3.8 inverses and radical Functions 303 How To… Given a ...
and y, this gave us the values y could assume. For this function, x β‰₯ 4, so for the inverse, we should have y β‰₯ 4, which is what our inverse function gives. a. The domain of the original function was restricted to x β‰₯ 4, so the outputs of the inverse need to be the same, f (x) β‰₯ 4, and we must use the + case: f βˆ’1(x) ...
x β‰₯ 2. To find the inverse, we will use the vertex form of the quadratic. We start by replacing f (x) with a simple variable, y, then solve for x. y = (x βˆ’ 2)2 βˆ’ 3 Interchange x and y. x = (y βˆ’ 2)2 βˆ’ 3 Add 3 to both sides. Take the square rooty βˆ’ 2)1(x Add 2 to both sides. Rename the function. Now we need to determine...
inverse of a radical function, we will need to restrict the domain of the answer because the range of the original function is limited. How To… Given a radical function, find the inverse. 1. Determine the range of the original function. 2. Replace f (x) with y, then solve for x. 3. If necessary, restrict the domain of...
r β‰₯ 0 since negative radii would not make sense in this context. Also note the range of the function (hence, the domain of the inverse function) is V β‰₯ 0. Solve for r in terms of V, using the method outlined previously. Ο€r 3 V = 2 __ 3 r 3 = 3V ___ 2Ο€ ____ √ 3V ___ 2Ο€ r = 3 Solve for r 3. Solve for r. This is the resu...
; this behavior is similar to the basic reciprocal toolkit function, and there is no horizontal asymptote because the degree of the numerator is larger than the degree of the denominator. There is a y-intercept at (0, √ β€” 6 ). From the y-intercept and x-intercept at x = βˆ’2, we can sketch the left side of the graph. Fro...
practice with inverses and radical functions. β€’ Graphing the Basic Square Root Function (http://openstaxcollege.org/l/graphsquareroot) β€’ Find the Inverse of a Square Root Function (http://openstaxcollege.org/l/inversesquare) β€’ Find the Inverse of a Rational Function (http://openstaxcollege.org/l/inverserational) β€’ Fin...
βˆ’ 4 20. f (x) = √ β€” 6x βˆ’ 8 + 5 21. f (x) = 9 + 2 3 √ β€” x 23. f (x) = 2 _____ x + 8 26. f (x) = x βˆ’ 2 _____ x + 7 24. f (x) = 3 _____ x βˆ’ 4 27. f (x) = 3x + 4 ______ 5 βˆ’ 4x 22. f (x) = 3 βˆ’ 3 √ β€” x 25. f (x) = x + 3 _____ x + 7 28. f (x) = 5x + 1 ______ 2 βˆ’ 5x 29. f (x) = x 2 + 2x, [βˆ’1, ∞) 30. f (x) = x 2 + 4x + 1, [βˆ’2,...
exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given. 47. f (x) = x3 βˆ’ x βˆ’ 2, y = 1, 2, 3 48. f (x) = x3 + x βˆ’ 2, y = 0, 1, 2 49. f (x) = x3 + 3x βˆ’ 4, y = 0, 1, 2 50. f (x) = x3 + 8x βˆ’ 4, y = βˆ’1, 0, 1 51. f (x) = x4 + 5x + 1, y...
solution that is 25 ml acid. If n ml of a solution that is 60% acid is added, the function C(n) = gives the 25 + 0.6n ________ 100 + n concentration, C, as a function of the number of ml added, n. Express n as a function of C and determine the number of mL that need to be added to have a solution that is 50% acid. 62....
600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate. Solving Direct Variation Problems In the example above, Nicole’s earnings can be found by multiplying her sales by her commis...
of the form y = kx n then we say that the relationship is direct variation and y varies directly with the nth power of x. In direct y _ variation relationships, there is a nonzero constant ratio k = xn, where k is called the constant of variation, which help defines the relationship between the variables. How To… Give...
28Β°F. If we create Table 2, we observe that, as the depth increases, the water temperature decreases. d, depth 500 ft 350 ft 250 ft T = 14,000 _ d 14,000 _ 500 = 28 14,000 _ 350 = 40 14,000 _ 250 = 56 Interpretation At a depth of 500 ft, the water temperature is 28Β° F. At a depth of 350 ft, the water temperature is 40...
, x, and the output, y. 2. Determine the constant of variation. You may need to multiply y by the specified power of x to determine the constant of variation. 3. Use the constant of variation to write an equation for the relationship. 4. Substitute known values into the equation to find the unknown. Example 3 Solving a...