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Sullivan Algebra and Trigonometry: Section 4.3 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

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A quadratic function is a function of the form: 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.

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Graphs of a quadratic function f(x) = ax 2 + bx + c, a 0 a > 0 Opens up Vertex is lowest point Axis of symmetry a < 0 Opens down Vertex is highest point Axis of symmetry

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Graph the function fxxx() 281 2 Find the vertex and axis of symmetry. First, rewrite the function by completing the square

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Now, graph the function using the transformations discussed in Chapter 3. (0,0) (2,4) (0,0) (2, -8)

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(2, 0) (4, -8) (2, 7) (4, -1) Vertex : (2,7)

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Properties of the Quadratic Function Parabola opens up if a > 0. Parabola opens down if a < 0.

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Given the function, determine whether the graph opens upward or downward. Find the vertex, axis of symmetry, the x- intercepts, and the y-intercept. fxxx() 2125 2 x-coord. of vertex: y-coord. of vertex: Axis of symmetry: x b a 2 3 Vertex: (-3, -13)

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fxxx() 2125 2 y-intercepts: f(0) = 5; so the y-intercept is (0,5) x-intercepts: Solve the equation = 0 fxxx() 2125 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)

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Now, graph the function using the information found in the previous steps. Vertex: (-3, -13) (-5.55, 0)(-0.45, 0) (0, 5)

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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 what unit price p should be charged to maximize revenue? What is the maximum revenue?

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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: y-coord. of vertex: So, the maximum revenue occurs when they charge $1900. The maximum revenue is $1,805,000.

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Suppose you throw a ball straight up and record the height of the ball in 0.5 second intervals and obtain the following data:

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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 Their Properties

Quadratic Functions and Their Properties

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