204 CHAPTER 4 Polynomial and Rational Functions 88. Mixed Practice Consider the function ( )=− + + f x x x2 3. 4 2 (a) Determine the maximum number of turning points on the graph of f. (b) Graph f using a graphing utility with window settings 5, 5, 1, 10, 10, 1 . [ ] − − Verify that the graph has the maximum number of turning points found in part (a). (c) Determine the end behavior of f; that is, find the power function that the graph of f resembles for large values of x . (d) Based on the results of parts (b) and (c), explain why the graph of f will not have any additional turning points off the viewing window. (e) The function f is decreasing where its derivative f x x x 4 4 0. 3 ( ) ′ = − + ≤ Use the derivative to determine the intervals for which f is decreasing. Because polynomials are continuous over their domain, all endpoints are included in the interval describing increasing/decreasing. However, in general, the numbers at the endpoints must be tested separately to determine if they should be included in the interval describing where a function is increasing or decreasing. (f) Use a graphing utility to determine the intervals for which f is decreasing to confirm your results from part (e). 85. Mixed Practice G x x x 3 2 2 ( ) ( ) ( ) = + − (a) Identify the x-intercepts of the graph of G. (b) What are the x-intercepts of the graph of y G x 3 ? ( ) = + In Problems 81–84, write a polynomial function whose graph is shown (use the smallest degree possible). 81. (1, –8) x y 10 –14 –6 6 82. (3, 8) x y 14 –10 –6 6 83. (2, –50) x 72 –72 –6 6 y 84. (–2, 16) x –2 2 21 –15 y 86. Mixed Practice h x x x 2 4 3 ( ) ( )( ) = + − (a) Identify the x-intercepts of the graph of h. (b) What are the x-intercepts of the graph of y h x 2 ? ( ) = − 87. Mixed Practice Consider the function ( )= − + f x x x 2 3 4. 3 2 (a) Determine the maximum number of turning points on the graph of f. (b) Graph f using a graphing utility with window settings 5, 5, 1, 10, 10, 1 . [ ] − − Verify that the graph has the maximum number of turning points found in part (a). (c) Determine the end behavior of f; that is, find the power function that the graph of f resembles for large values of x . (d) Based on the results of parts (b) and (c), explain why the graph of f will not have any additional turning points off the viewing window. (e) The function f is increasing where its derivative f x x x 6 6 0. 2 ( ) ′ = − ≥ Use the derivative to determine the intervals for which f is increasing. Because polynomials are continuous over their domain, all endpoints are included in the interval describing increasing/decreasing. However, in general, the numbers at the endpoints must be tested separately to determine if they should be included in the interval describing where a function is increasing or decreasing. (f) Use a graphing utility to determine the intervals for which f is increasing to confirm your results from part (e). 89. Challenge Problem Find the real zeros of f x x x x 3 1 4 3 2 2 2 ( )( ) ( ) = − + + and their multiplicity. 90. Challenge Problem Determine the power function that resembles the end behavior of g x x x x x 4 4 5 2 3 1 2 1 2 2 3 ( ) ( ) ( ) ( ) = − − − + Explaining Concepts 91. Can the graph of a polynomial function have no y-intercept? Can it have no x-intercepts? Explain. 92. The illustration shows the graph of a polynomial function. x y (a) Is the degree of the polynomial even or odd? (b) Is the leading coefficient positive or negative? (c) Is the function even, odd, or neither? (d) Why is x2 necessarily a factor of the polynomial? (e) What is the minimum degree of the polynomial? (f) Formulate five different polynomials whose graphs could look like the one shown. Compare yours to those of other students. What similarities do you see? What differences? 93. Which of the following statements are true regarding the graph of the polynomial function ( )= + + + f x x bx cx d? 3 2 (Give reasons for your conclusions.) (a) It intersects the y-axis in one and only one point. (b) It intersects the x-axis in at most three points. (c) It intersects the x-axis at least once. (d) For x very large, it behaves like the graph of y x .3 = (e) It is symmetric with respect to the origin. (f) It passes through the origin. 94. The graph of a polynomial function is always smooth and continuous. Name a function studied earlier that is smooth but not continuous. Name one that is continuous but not smooth. 95. Make up two polynomial functions, not of the same degree, with the following characteristics: crosses the x-axis at 2, − touches the x-axis at 1, and is above the x-axis between 2− and 1. Give your polynomials to a fellow classmate and ask for a written critique.
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