SECTION 3.3 Quadratic Functions and Their Properties 157 Retain Your Knowledge Problems 29–38 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. 29. Find an equation for the line containing the points ( ) −1, 5 and ( ) − 3, 3 . Write the equation using either the general form or the slope–intercept form, whichever you prefer. 30. Find the domain of ( ) = − − f x x x 1 25 . 2 31. For ( ) = − f x x5 8 and ( ) = − + g x x x3 4, 2 find ( )( ) −g f x . 32. Find a function whose graph is the graph of = y x ,2 but shifted to the left 3 units and shifted down 4 units. 33. Solve: − = x x4 3 2 34. Solve. Write the answer in interval notation. ( ) ( ) + − ≥ − + x x x 5 2 7 6 10 3 9 35. Determine algebraically whether the function ( ) = − f x x x 5 2 2 2 is even, odd, or neither. 36. Find the x -intercept(s) and y -intercept(s) of the graph of − = x y 3 8 6. 37. Rationalize the numerator: − + − ≥− ≠ x x x x 6 1 35 1, 35 38. Write ( ) ( ) + + + >− x x x x 4 2 3 2 , 2, 1 2 1 2 as a single quotient in which only positive exponents appear. ‘Are You Prepared?’ Answers 1. x y 3 6 9 12 1 2 3 Not a function because the input, 1, corresponds to two different outputs. 2. = + y x2 2 Quadratic Functions Here are some examples of quadratic functions. ( ) ( ) ( ) ( ) = = − + = − + = + f x x F x x x g x x H x x x 3 5 1 6 1 1 2 2 3 2 2 2 2 3.3 Quadratic Functions and Their Properties Now Work the ‘Are You Prepared?’ problems on page 166. • Intercepts ( Section 1.3 , pp. 20 – 21 ) • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) • Completing the Square (Section A.3, p. A29) • Quadratic Equations (Section A.6, pp. A47–A53) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Graph a Quadratic Function Using Transformations (p. 158) 2 Identify the Vertex and Axis of Symmetry of a Parabola (p. 160) 3 Graph a Quadratic Function Using Its Vertex, Axis, and Intercepts (p. 161) 4 Find a Quadratic Function Given Its Vertex and One Other Point (p. 164) 5 Find the Maximum or Minimum Value of a Quadratic Function (p. 164) DEFINITION Quadratic Function A quadratic function is a function of the form f x ax bx c 2 ( ) = + + where a , b , and c are real numbers and a 0. ≠ The domain of a quadratic function is the set of all real numbers.
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