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4.1 and 4.7 Graphing Quadratic Functions

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Quadratic function a function that has the form y = ax 2 + bx + c, where a cannot = 0.

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Parabola the graph of a quadratic function

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Vertex

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Axis of symmetry the vertical line through the vertex of a quadratic function A. O. S.

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Standard form when a quadratic function is written in the form: y = ax 2 + bx + c

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Graphing in standard form The graph of y = ax 2 + bx + c is a parabola with these characteristics opens up if a > 0 opens down if a < 0 The x-coordinate of the vertex is The axis of symmetry is the vertical line

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Example 1: Graph f(x) = –x 2 + x + 22. Then determine the domain and range of the function? xy 20 022 1/222.25 122 220 1. Find the vertex and AOS using –b/2a 2. Make a table with the vertex at the center. xy 3. Graph.

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Example 2 xy -4.5 0-6 1-6.5 2-6 3-4.5 Vertex (1, -6.5) AOS: x = 1 b) Find the domain and range of the function.

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Example 3: Determine whether the function has a maximum or a minimum value. Then find the value. a) f(x) = x 2 + 4x – 1 b) g(x) = -2x 2 + x + 8

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Vertex Form of a Quadratic Function Vertex form: y = a(x – h) 2 + k The vertex is (h, k). The axis of symmetry is x = h. The graph opens up if a > 0 and opens down if a < 0.

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Example 4: a) Graph f(x) = 2(x – 1) 2 + 3 xy 11 05 13 25 3 1. Identify the vertex. (1, 3) 2. Make a table with the vertex at the center. xy 0 13 2 3 b) Find the domain and range of the function.

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Example 5 Write the equation of the function shown in the graph.

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Example 6 A manufacturer of lighting fixtures has daily production costs modeled by y = 0.25x 2 – 10x + 800 where y is the total cost in dollars and x is the number of fixtures produced. What is the minimum daily production cost? How many fixtures should be produced each day to yield a minimum cost?

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Example 7 The path of a diver is approximated by feet in the figure shown and the equation given. Time is measured in seconds. What is the maximum height of the diver? Approximately how long did it take the diver to reach his maximum height?

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