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3.3E: Exercises

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Dec 21, 2020
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- 3.3: Power Functions and Polynomial Functions
- 3.4: Graphs of Polynomial Functions

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Section Exercises

Verbal

1. Explain the difference between the coefficient of a power function and its degree.

Answer: The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

2. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

Answer: As $x$ decreases without bound, so does $f(x)$ . As $x$ increases without bound, so does $f(x)$ .

4. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

5. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x→−∞, $f(x)→−∞$ and as x→∞, $f(x)→−∞$ .

Answer: The polynomial function is of even degree and leading coefficient is negative.

Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

6. $f(x)=x^5$

7. $f(x)=(x^2)^3$

Answer: Power function

8. $f(x)=x−x^4$

Exercise $\PageIndex{9}$

$f(x)=\frac{x^2}{x^2−1}$

Answer: Neither

10. $f(x)=2x(x+2)(x−1)^2$

11. $f(x)=3^{x+1}$

Answer: Neither

For the following exercises, find the degree and leading coefficient for the given polynomial.

12. $−3x^4$

13. $7−2x^2$

Answer: Degree = 2, Coefficient = –2

14. $−2x^2− 3x^5+ x−6$

15. $x(4−x^2)(2x+1)$

Answer: Degree =4, Coefficient = –2

16. $x^2(2x−3)^2$

For the following exercises, determine the end behavior of the functions.

17. $f(x)=x^4$

Answer: As $x→∞$ , $f(x)→∞$ , as $x→−∞$ , $f(x)→∞$

18. $f(x)=x^3$

19. $f(x)=−x^4$

Answer: As $x→−∞$ , $f(x)→−∞$ , as $x→∞$ , $f(x)→−∞$

20. $f(x)=−x^9$

21. $f(x)=−2x^4− 3x^2+ x−1$

Answer: As $x→−∞$ , $f(x)→−∞$ , as $x→∞$ , $f(x)→−∞$

22. $f(x)=3x^2+ x−2$

23. $f(x)=x^2(2x^3−x+1)$

Answer: As $x→∞$ , $f(x)→∞$ , as $x→−∞$ , $f(x)→−∞$

24. $f(x)=(2−x)^7$

For the following exercises, find the intercepts of the functions.

25. $f(t)=2(t−1)(t+2)(t−3)$

Answer: y-intercept is $(0,12)$ , t-intercepts are $(1,0);(–2,0);$ and $(3,0)$ .

26. $g(n)=−2(3n−1)(2n+1)$

Exercise $\PageIndex{27}$

$f(x)=x^4−16$

Answer: y-intercept is $(0,−16).$ x-intercepts are $(2,0)$ and $(−2,0)$ .

28. $f(x)=x^3+27$

29. $f(x)=x(x^2−2x−8)$

Answer: y-intercept is $(0,0)$ . x-intercepts are $(0,0),(4,0),$ and $(−2, 0)$ .

30. $f(x)=(x+3)(4x^2−1)$

Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.

31.

$CNX_Precalc_Figure_03_03_201.jpg$

Answer: 3

32.

$CNX_Precalc_Figure_03_03_202.jpg$

33.
$CNX_Precalc_Figure_03_03_203.jpg$

Answer: 5

34.

$CNX_Precalc_Figure_03_03_204.jpg$

35.
$CNX_Precalc_Figure_03_03_205.jpg$

Answer: 3

36.

$CNX_Precalc_Figure_03_03_206.jpg$

37.
$CNX_Precalc_Figure_03_03_207.jpg$

Answer: 5

38.

$CNX_Precalc_Figure_03_03_208.jpg$

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

39.

$CNX_Precalc_Figure_03_03_209.jpg$

Answer: Yes. Number of turning points is 2. Least possible degree is 3.

40.

$CNX_Precalc_Figure_03_03_210.jpg$

41.
$CNX_Precalc_Figure_03_03_211.jpg$

Answer: Yes. Number of turning points is 1. Least possible degree is 2.

42.

$CNX_Precalc_Figure_03_03_212.jpg$

43.
$CNX_Precalc_Figure_03_03_213.jpg$

Answer: Yes. Number of turning points is 0. Least possible degree is 3.

Exercise $\PageIndex{44}$

$CNX_Precalc_Figure_03_03_214.jpg$

Answer: No (the graph is not smooth)

45.

$CNX_Precalc_Figure_03_03_215.jpg$

Answer: Yes. Number of turning points is 0. Least possible degree is 1.

Numeric

For the following exercises, make a table to confirm the end behavior of the function.

46. $f(x)=−x^3$

47. $f(x)=x^4−5x^2$

Answer:

$x$	$f(x)$
10	9,500
100	99,950,000
–10	9,500
–100	99,950,000

as $x→−∞,$ $f(x)→∞$ , as $x→∞,$ $f(x)→∞$

48. $f(x)=x^2(1−x)^2$

49. $f(x)=(x−1)(x−2)(3−x)$

Answer:

$x$	$f(x)$
10	9,500
100	99,950,000
–10	9,500
–100	99,950,000

as $x→−∞,$ $f(x)→∞$ , as $x→∞,$ $f(x)→−∞$

50. $f(x)=\frac{x^5}{10}−x^4$

Technology

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

51. $f(x)=x^3(x−2)$

Answer:

$CNX_Precalc_Figure_03_03_216.jpg$

The y-intercept is $(0, 0)$ . The x-intercepts are $(0, 0), (2, 0).$ As $x→−∞,$ $f(x)→∞$ , as $x→∞,$ $f(x)→∞$

52. $f(x)=x(x−3)(x+3)$

53. $f(x)=x(14−2x)(10−2x)$

Answer:

$CNX_Precalc_Figure_03_03_218.jpg$

The y-intercept is $(0,0)$ . The x-intercepts are $(0, 0), (5, 0), (7, 0)$ . As $x→−∞$ , $f(x)→−∞$ , as $x→∞,$ $f(x)→∞$

54. $f(x)=x(14−2x)(10−2x)^2$

55. $f(x)=x^3−16x$

Answer:

$CNX_Precalc_Figure_03_03_220.jpg$

The y-intercept is (0, 0). The x-intercept is $(−4, 0), (0, 0), (4, 0)$ . As $x→−∞$ , $f(x)→−∞$ , as $x→∞,$ $f(x)→∞$

56. $f(x)=x^3−27$

57. $f(x)=x^4−81$

Answer:

$CNX_Precalc_Figure_03_03_222.jpg$

The y-intercept is (0, −81). The x-intercept are $(3, 0), (−3, 0)$ . As $x→−∞,$ $f(x)→∞$ , as $x→∞,$ $f(x)→∞$

58. $f(x)=−x^3+x^2+2x$

59. $f(x)=x^3−2x^2−15x$

Answer:

$CNX_Precalc_Figure_03_03_224.jpg$

The y-intercept is $(0, 0)$ . The x-intercepts are $(−3, 0), (0, 0), (5, 0).$ As $x→−∞$ , $f(x)→−∞$ , as $x→∞,$ $f(x)→∞$

60. $f(x)=x^3−0.01x$

Extensions

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

61. The y-intercept is $(0,−4)$ . The x-intercepts are $(−2,0), (2,0)$ . Degree is 2.

End behavior: as $x→−∞,$ $f(x)→∞$ , as $x→∞,$ $f(x)→∞$ .

Answer: $f(x)=x^2−4$

62. The y-intercept is $(0,9)$ . The x-intercepts are $(−3,0), (3,0)$ . Degree is 2.

End behavior: as $x→−∞,$ $f(x)→−∞$ , as $x→∞,$ $f(x)→−∞$ .

63. The y-intercept is $(0,0)$ . The x-intercepts are $(0,0), (2,0)$ . Degree is 3.

End behavior: as $x→−∞,$ $f(x)→−∞$ , as $x→∞,$ $f(x)→∞$ .

Answer: $f(x)=x^3−4x^2+4x$

64. The y-intercept is $(0,1)$ . The x-intercept is $(1,0)$ . Degree is 3.

End behavior: as $x→−∞$ , $f(x)→∞$ , as $x→∞$ , $f(x)→−∞$ .

65. The y-intercept is $(0,1)$ . There is no x-intercept. Degree is 4.

End behavior: as $x→−∞,$ $f(x)→∞$ , as $x→∞,$ $f(x)→∞$ .

Answer: $f(x)=x^4+1$

Real-World Applications

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

66. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of $d$ , the number of days elapsed.

67. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of $m$ , the number of minutes elapsed.

Answer: $V(m)=8m^3+36m^2+54m+27$

68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by $x$ inches and the width increased by twice that amount, express the area of the rectangle as a function of $x$ .

Exercise $\PageIndex{69}$

An open box is to be constructed by cutting out square corners of $x$ -inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of $x$ .

Answer: $V(x)=4x^3−32x^2+64x$

70. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width (x).

Section Exercises

Verbal

Algebraic

Graphical

Numeric

Technology

Extensions

Real-World Applications

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