SECTION 4.2 The Graph of a Polynomial Function; Models 205 98. Design a polynomial function with the following characteristics: degree 6; four distinct real zeros, one of multiplicity 3; y -intercept 3; behaves like y x5 6 = − for large values of x . Is this polynomial unique? Compare your polynomial with those of other students. What terms will be the same as everyone else’s? Add some more characteristics, such as symmetry or naming the real zeros. How does this modify the polynomial? 96. Make up a polynomial function that has the following characteristics: crosses the x -axis at 1− and 4, touches the x - axis at 0 and 2, and is above the x -axis between 0 and 2. Give your polynomial to a fellow classmate and ask for a written critique. 97. Write a few paragraphs that provide a general strategy for graphing a polynomial function. Be sure to mention the following: degree, intercepts, end behavior, and turning points. Retain Your Knowledge Problems 99–107 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus. 99. Find an equation of the line that contains the point 2, 3 ( ) − and is perpendicular to the line x y 5 2 6. − = 100. Find the domain of the function h x x x 3 5 . ( ) = − + 101. Use the quadratic formula to find the real zeros of the function f x x x 4 8 3. 2 ( ) = + − 102. Solve: − = x5 3 7 103. Determine whether the function f x x3 2 ( ) = − + is increasing, decreasing, or constant. 104. Find the function that is finally graphed if the graph of f x x2 ( ) = is shifted left 2 units and up 5 units. 105. Find the difference quotient of f x x 1 3 2. ( ) = − + 106. The midpoint of a line segment is 3, 5 ( ) − and one endpoint is 2, 4 . ( ) − Find the other endpoint. 107. Find the quotient and remainder if x x 4 7 5 3 2 − + is divided by x 1. 2 − 108. The function f x x x 2 3 4 3 2 ( ) = − + is increasing where its derivative f x x x 6 6 0. 2 ( ) ′ = − > Where is f increasing? Verify your result by graphing f using a graphing utility. ‘Are You Prepared?’ Answers 1. 2, 0 , 2, 0 , 0, 9 ( ) ( ) ( ) − 2. Yes; 3 3. Down; 4 4. True 5. b; c 4.2 The Graph of a Polynomial Function; Models Now Work the ‘Are You Prepared?’ problems on page 211. • Local Maxima and Local Minima (Section 2.3, pp. 90–91) • Using a Graphing Utility to Approximate Local Maxima and Local Minima (Section 2.3, p. 93) • Intercepts (Section 1.3, pp. 20–21) • Building Quadratic Models from Data (Section 3.4, pp. 174–175) • Polynomials (Section A.3, pp. A23–A29) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Analyze the Graph of a Polynomial Function (p. 206) 2 Build Cubic Models from Data (p. 210) We now use the properties discussed in the previous section to analyze graphs of polynomial functions.
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