4.1 Section Exercises
Verbal
1. Explain the difference between the coefficient of a power function and its degree.
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.
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[latex]\text{ }x\to -\infty ,\text{ }f\left(x\right)\to -\infty \text{ }[/latex] and as[latex]\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty .\text{ }[/latex]
Algebraic
For the following exercises, identify the function as a power function, a polynomial function, or neither.
6. [latex]f\left(x\right)={x}^{5}[/latex]
7. [latex]f\left(x\right)={\left({x}^{2}\right)}^{3}[/latex]
8. [latex]f\left(x\right)=x-{x}^{4}[/latex]
9. [latex]f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-1}[/latex]
10. [latex]f\left(x\right)=2x\left(x+2\right){\left(x-1\right)}^{2}[/latex]
11. [latex]f\left(x\right)={3}^{x+1}[/latex]
For the following exercises, find the degree and leading coefficient for the given polynomial.
12. [latex]-3x{}^{4}[/latex]
13. [latex]7-2{x}^{2}[/latex]
14. [latex]-2{x}^{2}- 3{x}^{5}+ x-6[/latex]
15. [latex]x\left(4-{x}^{2}\right)\left(2x+1\right)[/latex]
16. [latex]{x}^{2}{\left(2x-3\right)}^{2}[/latex]
For the following exercises, determine the end behavior of the functions.
17. [latex]f\left(x\right)={x}^{4}[/latex]
18. [latex]f\left(x\right)={x}^{3}[/latex]
19. [latex]f\left(x\right)=-{x}^{4}[/latex]
20. [latex]f\left(x\right)=-{x}^{9}[/latex]
21. [latex]f\left(x\right)=-2{x}^{4}- 3{x}^{2}+ x-1[/latex]
22. [latex]f\left(x\right)=3{x}^{2}+ x-2[/latex]
23. [latex]f\left(x\right)={x}^{2}\left(2{x}^{3}-x+1\right)[/latex]
24. [latex]f\left(x\right)={\left(2-x\right)}^{7}[/latex]
For the following exercises, find the intercepts of the functions.
25. [latex]f\left(t\right)=2\left(t-1\right)\left(t+2\right)\left(t-3\right)[/latex]
26. [latex]g\left(n\right)=-2\left(3n-1\right)\left(2n+1\right)[/latex]
27. [latex]f\left(x\right)={x}^{4}-16[/latex]
28. [latex]f\left(x\right)={x}^{3}+27[/latex]
29. [latex]f\left(x\right)=x\left({x}^{2}-2x-8\right)[/latex]
30. [latex]f\left(x\right)=\left(x+3\right)\left(4{x}^{2}-1\right)[/latex]
Graphical
For the following exercises, determine the least possible degree of the polynomial function shown.
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.
Numeric
For the following exercises, make a table to confirm the end behavior of the function.
46. [latex]f\left(x\right)=-{x}^{3}[/latex]
47. [latex]f\left(x\right)={x}^{4}-5{x}^{2}[/latex]
[latex]\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty[/latex]
48. [latex]f\left(x\right)={x}^{2}{\left(1-x\right)}^{2}[/latex]
49. [latex]f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(3-x\right)[/latex]
50. [latex]f\left(x\right)=\frac{{x}^{5}}{10}-{x}^{4}[/latex]
Technology
For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.
51. [latex]f\left(x\right)={x}^{3}\left(x-2\right)[/latex]
52. [latex]f\left(x\right)=x\left(x-3\right)\left(x+3\right)[/latex]
53. [latex]f\left(x\right)=x\left(14-2x\right)\left(10-2x\right)[/latex]
54. [latex]f\left(x\right)=x\left(14-2x\right){\left(10-2x\right)}^{2}[/latex]
55. [latex]f\left(x\right)={x}^{3}-16x[/latex]
56. [latex]f\left(x\right)={x}^{3}-27[/latex]
57. [latex]f\left(x\right)={x}^{4}-81[/latex]
58. [latex]f\left(x\right)=-{x}^{3}+{x}^{2}+2x[/latex]
59. [latex]f\left(x\right)={x}^{3}-2{x}^{2}-15x[/latex]
60. [latex]f\left(x\right)={x}^{3}-0.01x[/latex]
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[latex]\text{ }y-[/latex]intercept is[latex]\text{ }\left(0,-4\right).\text{ }[/latex]The[latex]\text{ }x-[/latex]intercepts are[latex]\text{ }\left(-2,0\right),\text{ }\left(2,0\right).\text{ }[/latex]Degree is 2.
End behavior:[latex]\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty .[/latex]
62. The[latex]\text{ }y-[/latex]intercept is[latex]\text{ }\left(0,9\right).\text{ }[/latex]The[latex]\text{ }x\text{-}[/latex]intercepts are[latex]\text{ }\left(-3,0\right),\text{ }\left(3,0\right).\text{ }[/latex]Degree is 2.
End behavior:[latex]\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty .[/latex]
63. The[latex]\text{ }y-[/latex]intercept is[latex]\text{ }\left(0,0\right).\text{ }[/latex]The[latex]\text{ }x-[/latex]intercepts are[latex]\text{ }\left(0,0\right),\text{ }\left(2,0\right).\text{ }[/latex]Degree is 3.
End behavior:[latex]\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty .[/latex]
64. The[latex]\text{ }y-[/latex]intercept is[latex]\text{ }\left(0,1\right).\text{ }[/latex]The[latex]\text{ }x-[/latex]intercept is[latex]\text{ }\left(1,0\right).\text{ }[/latex]Degree is 3.
End behavior:[latex]\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty .[/latex]
65. The[latex]\text{ }y-[/latex]intercept is[latex]\text{ }\left(0,1\right).\text{ }[/latex]There is no[latex]\text{ }x-[/latex]intercept. Degree is 4.
End behavior:[latex]\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty .[/latex]
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[latex]\text{ }d,\text{ }[/latex]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[latex]\text{ }m,\text{ }[/latex]the number of minutes elapsed.
68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by[latex]\text{ }x\text{ }[/latex]inches and the width increased by twice that amount, express the area of the rectangle as a function of[latex]\text{ }x.[/latex]
69. An open box is to be constructed by cutting out square corners of [latex]\text{ }x-[/latex]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[latex]\text{ }x.[/latex]
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 ([latex]x[/latex]).
