170 CHAPTER 3 Linear and Quadratic Functions derivative of f is given by f x x x 3 14 5. 2 ( ) ′ = − − The function f is increasing where f x 0 ( ) ′ > and decreasing where f x 0. ( ) ′ < 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 and decreasing. Because polynomials are continuous over its domain, all endpoints are included in the interval describing increasing/decreasing. Determine where f is increasing and where f is decreasing. 104. Challenge Problem Test for Concavity Suppose f x x x x 3 8 6 1. 4 3 ( ) = − + + From calculus, the second derivative of f is given by ( ) ″ = − f x x x 36 48 . 2 The function f is concave up where ( ) ″ > f x 0 and concave down where ( ) ″ < f x 0. Determine where f is concave up and where f is concave down. 98. Mixed Practice Find the distance from the vertex of the parabola f x x 2 3 5 2 ( ) ( ) = − + to the center of the circle x y 3 1 4. 2 2 ( ) ( ) + + − = 99. Mixed Practice Find the distance from the vertex of the parabola g x x x 3 6 1 2 ( ) = − + + to the center of the circle x y x y 10 8 32 0. 2 2 + + + + = 100. Challenge Problem Let f x ax bx c, 2 ( ) = + + where a, b, and c are odd integers. If x is an integer, show that f x( ) must be an odd integer. [Hint: x is either an even integer or an odd integer.] 101. Challenge Problem Find the point on the line y x = that is closest to the point 3, 1 ( ). 102. Challenge Problem Find the point on the line y x 1 = + that is closest to the point 4, 1 ( ). 103. Challenge Problem Increasing/Decreasing Function Test Suppose f x x x x 7 5 35. 3 2 ( ) = − − + From calculus, the ‘Are You Prepared?’ Answers 1. 0, 9, 3,0, 3,0 ( ) ( ) ( ) − − 2. 4, 1 2 { } − 3. 25 4 4. right; 4 5. 89; two real solutions 6. ( ) ( ) + + = + x x x 3 7 3 49 36 3 7 6 2 2 Explaining Concepts 105. Make up a quadratic function that opens down and has only one x-intercept. Compare yours with others in the class. What are the similarities? What are the differences? 106. On one set of coordinate axes, graph the family of parabolas f x x x c 2 2 ( ) = + + for c c 3, 0, = − = and c 1. = Describe the characteristics of a member of this family. 107. On one set of coordinate axes, graph the family of parabolas f x x bx 1 2 ( ) = + + for b b 4, 0, = − = and b 4. = Describe the general characteristics of this family. 108. State the circumstances that cause the graph of a quadratic function f x ax bx c 2 ( ) = + + to have no x-intercepts. 109. Why is the graph of a quadratic function concave up if a 0 > and concave down if a 0? < 110. Can a quadratic function have a range of , ? ( ) −∞ ∞ Justify your answer. 111. What are the possibilities for the number of times the graphs of two different quadratic functions intersect? 112. In your own words, explain why the graph of a quadratic function will have no x-intercepts if the vertex lies in Quadrants III or IV and the parabola is concave down. Retain Your Knowledge Problems 113–122 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 subsequent sections, a final exam, or later courses such as calculus. 113. Determine whether x y4 16 2 2 + = is symmetric with respect to the x-axis, the y-axis, and/or the origin. 114. Solve the inequality x x 27 5 3. − ≥ + Write the solution in both set notation and interval notation. 115. Find the center and radius of the circle x y x y 10 4 20 0 2 2 + − + + = 116. Find the function whose graph is the graph of y x, = but reflected about the y-axis. 117. Find an equation of the line that contains the point 14, 3 ( ) − and is parallel to the line x y 5 7 35. + = Write the equation in slope-intercept form. 118. State the domain and range of the relation given below. Is the relation a function? 5, 3, 4, 4, 3, 5, 2, 6, 1, 7 ( ) ( ) ( ) ( ) ( ) { } − − − − − 119. If f x x x 3 25 28, 2 ( ) = − + find f 7( ). 120. If ( ) = − g x x 2 3 8, find ( ) + g x 3 2 12 . 121. Write x x x x 4 3 5 8 3 5 2 2/3 1/3 ( ) ( ) + + + as a single quotient with positive exponents. 122. If f x x x5 , 2 ( ) = + find and simplify f x f c x c x c , ( ) ( ) − − ≠
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