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Study Guides > MTH 163, Precalculus

Section Exercises

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 x,f(x)x\to -\infty ,f\left(x\right)\to -\infty \\ and as x,f(x)x\to \infty ,f\left(x\right)\to -\infty \\. For the following exercises, identify the function as a power function, a polynomial function, or neither. 6. f(x)=x5f\left(x\right)={x}^{5}\\ 7. f(x)=(x2)3f\left(x\right)={\left({x}^{2}\right)}^{3}\\ 8. f(x)=xx4f\left(x\right)=x-{x}^{4}\\ 9. f(x)=x2x21f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-1}\\ 10. f(x)=2x(x+2)(x1)2f\left(x\right)=2x\left(x+2\right){\left(x - 1\right)}^{2}\\ 11. f(x)=3x+1f\left(x\right)={3}^{x+1}\\ For the following exercises, find the degree and leading coefficient for the given polynomial. 12. 3x4-3x{}^{4}\\ 13. 72x27 - 2{x}^{2}\\ 14. 2x23x5+x6-2{x}^{2}- 3{x}^{5}+ x - 6 \\ 15. x(4x2)(2x+1)x\left(4-{x}^{2}\right)\left(2x+1\right)\\ 16. x2(2x3)2{x}^{2}{\left(2x - 3\right)}^{2}\\ For the following exercises, determine the end behavior of the functions. 17. f(x)=x4f\left(x\right)={x}^{4}\\ 18. f(x)=x3f\left(x\right)={x}^{3}\\ 19. f(x)=x4f\left(x\right)=-{x}^{4}\\ 20. f(x)=x9f\left(x\right)=-{x}^{9}\\ 21. f(x)=2x43x2+x1f\left(x\right)=-2{x}^{4}- 3{x}^{2}+ x - 1\\ 22. f(x)=3x2+x2f\left(x\right)=3{x}^{2}+ x - 2\\ 23. f(x)=x2(2x3x+1)f\left(x\right)={x}^{2}\left(2{x}^{3}-x+1\right)\\ 24. f(x)=(2x)7f\left(x\right)={\left(2-x\right)}^{7}\\ For the following exercises, find the intercepts of the functions. 25. f(t)=2(t1)(t+2)(t3)f\left(t\right)=2\left(t - 1\right)\left(t+2\right)\left(t - 3\right)\\ 26. g(n)=2(3n1)(2n+1)g\left(n\right)=-2\left(3n - 1\right)\left(2n+1\right)\\ 27. f(x)=x416f\left(x\right)={x}^{4}-16\\ 28. f(x)=x3+27f\left(x\right)={x}^{3}+27\\ 29. f(x)=x(x22x8)f\left(x\right)=x\left({x}^{2}-2x - 8\right)\\ 30. f(x)=(x+3)(4x21)f\left(x\right)=\left(x+3\right)\left(4{x}^{2}-1\right)\\ For the following exercises, determine the least possible degree of the polynomial function shown. 31. Graph of an odd-degree polynomial. 32. Graph of an even-degree polynomial. 33. Graph of an odd-degree polynomial. 34. Graph of an odd-degree polynomial. 35. Graph of an odd-degree polynomial. 36. Graph of an even-degree polynomial. 37. Graph of an odd-degree polynomial. 38. Graph of an even-degree polynomial. 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. Graph of an odd-degree polynomial. 40. Graph of an equation. 41. Graph of an even-degree polynomial. 42. Graph of an odd-degree polynomial. 43. Graph of an odd-degree polynomial. 44. Graph of an equation. 45. Graph of an odd-degree polynomial. For the following exercises, make a table to confirm the end behavior of the function. 46. f(x)=x3f\left(x\right)=-{x}^{3}\\ 47. f(x)=x45x2f\left(x\right)={x}^{4}-5{x}^{2}\\ 48. f(x)=x2(1x)2f\left(x\right)={x}^{2}{\left(1-x\right)}^{2}\\ 49. f(x)=(x1)(x2)(3x)f\left(x\right)=\left(x - 1\right)\left(x - 2\right)\left(3-x\right)\\ 50. f(x)=x510x4f\left(x\right)=\frac{{x}^{5}}{10}-{x}^{4}\\ 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(x2)f\left(x\right)={x}^{3}\left(x - 2\right)\\ 52. f(x)=x(x3)(x+3)f\left(x\right)=x\left(x - 3\right)\left(x+3\right)\\ 53. f(x)=x(142x)(102x)f\left(x\right)=x\left(14 - 2x\right)\left(10 - 2x\right)\\ 54. f(x)=x(142x)(102x)2f\left(x\right)=x\left(14 - 2x\right){\left(10 - 2x\right)}^{2}\\ 55. f(x)=x316xf\left(x\right)={x}^{3}-16x\\ 56. f(x)=x327f\left(x\right)={x}^{3}-27\\ 57. f(x)=x481f\left(x\right)={x}^{4}-81\\ 58. f(x)=x3+x2+2xf\left(x\right)=-{x}^{3}+{x}^{2}+2x\\ 59. f(x)=x32x215xf\left(x\right)={x}^{3}-2{x}^{2}-15x\\ 60. f(x)=x30.01xf\left(x\right)={x}^{3}-0.01x\\ 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)\left(0,-4\right)\\. The x-intercepts are (2,0),(2,0)\left(-2,0\right),\left(2,0\right)\\. Degree is 2. End behavior: as x,f(x), as x,f(x)\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty \\. 62. The y-intercept is (0,9)\left(0,9\right)\\. The x-intercepts are (3,0),(3,0)\left(-3,0\right),\left(3,0\right)\\. Degree is 2. End behavior: as x,f(x), as x,f(x)\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty\\ . 63. The y-intercept is (0,0)\left(0,0\right)\\. The x-intercepts are (0,0),(2,0)\left(0,0\right),\left(2,0\right)\\. Degree is 3. End behavior: as x,f(x), as x,f(x)\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty \\. 64. The y-intercept is (0,1)\left(0,1\right)\\. The x-intercept is (1,0)\left(1,0\right)\\. Degree is 3. End behavior: as x,f(x), as x,f(x)\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty \\. 65. The y-intercept is (0,1)\left(0,1\right)\\. There is no x-intercept. Degree is 4. End behavior: as x,f(x), as x,f(x)\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty\\. 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. 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. 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. 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).

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