Sullivan PreCalculus Section 3.1 Quadratic Functions/Models Objectives • Graph a Quadratic Function Using Transformations • Identify the Vertex and Axis of Symmetry of a Quadratic Function • Graph a Quadratic Function Using the Vertex, Axis of Symmetry, and Intercepts • Use a Graphing Utility to Find the Quadratic Function of Best Fit 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 consists of all real numbers. The graph of a quadratic function is called a parabola. Graphs of a quadratic function f(x) = ax2 + bx + c, a 0 Axis of symmetry Vertex is lowest point a>0 Opens up Vertex is highest point Axis of symmetry a<0 Opens down 2 f ( x ) 2 x - 8x - 1 Graph the function Find the vertex and axis of symmetry. First, rewrite the function by completing the square f ( x ) - 2 x 8 x - 1 - 2 x - 4 x - 1 2 2 -2 x - 4 x _ - 1 2 2 - 4 4 2 - 2 x - 4 x 4 - 1 8 2 - 2 x - 2 7 2 Now, graph the function using the transformations discussed in Chapter 3. 15 15 (2,4) (0,0) 10 (0,0) 0 10 10 0 10 (2, -8) 15 15 yx 2 y -2 x 2 15 15 (2, 7) (2, 0) 10 0 10 10 0 (4, -1)10 (4, -8) 15 y - 2 x - 2 15 2 y -2 x - 2 7 2 Vertex : (2,7) Properties of the Quadratic Function 2 f(x) ax bx c - b - b Vertex , f 2a 2a -b Axis of Symmetry : The line x 2a Parabola opens up if a > 0. Parabola opens down if a < 0. Given the function f ( x) 2x2 12x 5, determine whether the graph opens upward or downward. Find the vertex, axis of symmetry, the xintercepts, and the y-intercept. a = 2, b = 12, c = 5 a = 2 > 0, the parabola opens up x-coord. of vertex: - b - 12 -3 2a 22 -b y-coord. of vertex: f f -3 2(-3)2 12(-3) 5 -13 2a Vertex: (-3, -13) b x -3 Axis of symmetry: 2a 2 f ( x) 2x 12x 5 y-intercepts: f(0) = 5; so the y-intercept is (0,5) 2 x-intercepts: Solve the equation f ( x) 2x 12x 5 = 0 - b b 2 - 4ac x 2a - 12 104 2(2) 26 -3 2 The x-intercepts are approximately (-5.6,0) and (-.45,0) Summary: Parabola opens up Vertex (-3, -13) y-intercept: (0,5) x-intercepts: (-0.45, 0) and (-5.55,0) Now, graph the function using the information found in the previous steps. 10 (0, 5) (-5.55, 0) (-0.45, 0) 5 0 10 Vertex: (-3, -13) If a mathematical model of a real world situation leads to a quadratic function, the properties of the function can be applied to the model. Example: The John Deere Company has found that the revenue from sales of heavy duty tractors is a function of the unit price p that it charges. If the revenue R is -1 2 R( p) p 1900 p 2 what unit price p should be charged to maximize revenue? What is the maximum revenue? Note that the revenue function is a quadratic function that opens downward, since a < 0. So, the maximum revenue will be achieved at the vertex of the function. x-coord. of vertex: -b - 1900 1900 2a 2 - 1 2 y-coord. of vertex: -1 -b 2 f f 1900 1900 1900(1900) 1805000 2 2a So, the maximum revenue occurs when they charge $1900. The maximum revenue is $1,805,000. Suppose you throw a ball straight up and record the height of the ball in 0.5 second intervals and obtain the following data: time, t (in seconds) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 height, s (in feet) 51 89 120 141 155 162 160 149 133 106 73 29 Use a graphing utility to complete the following: a.) Draw a scatter diagram of the data. b.) Find the quadratic function of best fit. c.) Draw the quadratic function of best fit on the scatter diagram.

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# Quadratic Functions and Models