5.3.2: Homework

Reading Questions

1. What is the general form of a power function? Give an example of a toolkit function that is a power function.
2. Is \(f(x) = 2^x\) a power function? Why or why not?
3. How does the end behavior of an even-degree power function \(f(x)=kx^n\) (where \(n\) is even) differ based on the sign of \(k\)?
4. How does the end behavior of an odd-degree power function \(f(x)=kx^n\) (where \(n\) is odd) differ based on the sign of \(k\)?
5. What is the general form of a polynomial function? What restriction is placed on the exponents of the variable?
6. Define the "degree" of a polynomial.
7. Define the "leading term" and "leading coefficient" of a polynomial.
8. How does the leading term of a polynomial function relate to its end behavior?
9. What is a "turning point" of a polynomial graph?
10. What is the maximum number of \(x\)-intercepts a polynomial of degree \(n\) can have?
11. What is the maximum number of turning points a polynomial of degree \(n\) can have?
12. How do you find the \(y\)-intercept of a polynomial function given in general form?
13. If a polynomial function is given in factored form, like \(f(x)=(x-2)(x+1)(x-4)\), how can you find its \(x\)-intercepts?
14. Are the graphs of polynomial functions smooth and continuous? What do these terms mean informally?
15. When computing the difference quotient for \(f(x)=x^4\), what algebraic techniques might be useful for expanding \((x+h)^4\)?
16. Explain the difference between the coefficient of a power function and its degree.
17. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?
18. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.
19. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?
20. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As \(x \to − \infty\), \(f(x) \to − \infty\) and as \(x \to \infty\), \(f(x) \to − \infty \).

Homework

Algebraic

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

21. \(f(x)= x^5\)
22. \(f(x)= ( x^2 )^3\)
23. \(f(x)=x− x^4\)
24. \(f(x)= \frac{x^2}{x^2 −1}\)
25. \(f(x)=2x( x+2 ) ( x−1 )^2\)
26. \(f(x)= 3^{x+1}\)

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

27. \( -3x^4\)
28. \(7−2 x^2\)
29. \(−2 x^2 −3 x^5 +x−6\)
30. \(x( 4− x^2 )(2x+1)\)
31. \(x^2 ( 2x−3 )^2\)

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

32. \(f( x )= x^4\)
33. \(f( x )= x^3\)
34. \(f( x )=− x^4\)
35. \(f( x )=− x^9\)
36. \(f(x)=−2 x^4 −3 x^2 +x−1\)
37. \(f(x)=3 x^2 +x−2\)
38. \(f(x)= x^2 (2 x^3 −x+1)\)
39. \(f(x)= (2−x)^7\)

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

40. \(f( t )=2( t−1 )( t+2 )(t−3)\)
41. \(g( n )=−2( 3n−1 )(2n+1)\)
42. \(f(x)= x^4 −16\)
43. \(f(x)= x^3 +27\)
44. \(f(x)=x( x^2 −2x−8 )\)
45. \(f(x)=(x+3)(4 x^2 −1)\)

Graphical

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

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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.

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Numeric

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

61. \(f(x)=− x^3\)
62. \(f(x)= x^4 −5 x^2\)
63. \(f(x)= x^2 ( 1−x )^2\)
64. \(f(x)=(x−1)(x−2)(3−x)\)
65. \(f(x)= \frac{x^5}{10} − x^4\)

Technology

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

66. \(f(x)= x^3 (x−2)\)
67. \(f(x)=x(x−3)(x+3)\)
68. \(f(x)=x(14−2x)(10−2x)\)
69. \(f(x)=x(14−2x) (10−2x)^2\)
70. \(f(x)= x^3 −16x\)
71. \(f(x)= x^3 −27\)
72. \(f(x)= x^4 −81\)
73. \(f(x)=− x^3 + x^2 +2x\)
74. \(f(x)= x^3 −2 x^2 −15x\)
75. \(f(x)= x^3 −0.01x\)

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.

76. The \( y \)-intercept is \((0,−4)\). The \( x \)-intercepts are \((−2,0)\), \((2,0)\). Degree is 2. End behavior: as \(x \to − \infty\), \(f(x) \to \infty\), and as \(x \to \infty\), \(f(x) \to \infty \).
77. The \( y \)-intercept is \((0,9)\). The \( x \)-intercepts are \((−3,0)\), \((3,0)\). Degree is 2. End behavior: as \(x \to − \infty\), \(f(x) \to − \infty\), and as \(x \to \infty\), \(f(x) \to − \infty \).
78. The \( y \)-intercept is \((0,0)\). The \( x \)-intercepts are \((0,0)\), \((2,0)\). Degree is 3. End behavior: as \(x \to − \infty\), \(f(x) \to − \infty\), and as \(x \to \infty\), \(f(x) \to \infty \).
79. The \( y \)-intercept is \((0,1)\). The \( x \)-intercept is \((1,0)\). Degree is 3. End behavior: as \(x \to − \infty\), \(f(x) \to \infty\), and as \(x \to \infty\), \(f(x) \to − \infty \).
80. The \( y \)-intercept is \((0,1)\). There is no \( x \)-intercept. Degree is 4. End behavior: as \(x \to − \infty\), \(f(x) \to \infty\), and as \(x \to \infty\), \(f(x) \to \infty \).

Real-World Applications

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

81. 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.
82. 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.
83. 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 \(x\).
84. 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 \(x\).
85. 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 ( \(x\) ).