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Vocabulary quadratic function- a nonlinear function with an x squared term parabola- the graph of a quadratic function. It is a U-shaped graph. axis of symmetry- a central line that makes both sides of the parabola symmetric. vertex- highest or lowest point on the graph minimum- lowest point on the graph maximum- highest point on the graph

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Concept

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Example 1 Graph a Parabola Use a table of values to graph y = x 2 – x – 2. State the domain and range. Use a table of values to graph y = x 2 + 2x + 3. State the domain and range. A. B.

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Example 2 Identify Characteristics from Graphs A. Find the vertex, the equation of the axis of symmetry, and y-intercept of the graph. B. Find the vertex, the equation of the axis of symmetry, and y-intercept of the graph.

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Example 2 C. Consider the graph of y = 3x 2 – 6x + 1. Write the equation of the axis of symmetry, and find the coordinates of the vertex.

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Example 3 Identify Characteristics from Functions A. Find the vertex, the equation of the axis of symmetry, and y-intercept of y = –2x 2 – 8x – 2. B. Find the vertex, the equation of the axis of symmetry, and y-intercept of y = 3x 2 + 6x – 2. C. Find the vertex for y = x 2 + 2x – 3. D. Find the equation of the axis of symmetry for y = 7x 2 – 7x – 5.

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Example 4 Maximum and Minimum Values A. Consider f(x) = –x 2 – 2x – 2. Determine whether the function has a maximum or a minimum value. B. Consider f(x) = –x 2 – 2x – 2. State the maximum or minimum value of the function. C. Consider f(x) = –x 2 – 2x – 2. State the domain and range of the function. D. Consider f(x) = 2x 2 – 4x + 8. Determine whether the function has a maximum or a minimum value.

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Example 4 A.–1 B.1 C.6 D.8 E. Consider f(x) = 2x 2 – 4x + 8. State the maximum or minimum value of the function. F. Consider f(x) = 2x 2 – 4x + 8. State the domain and range of the function.

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Example 5 Graph Quadratic Functions A. Graph the function f(x) = –x 2 + 5x – 2. B. Graph the function f(x) = x 2 + 2x – 2.

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Example 6 Use a Graph of a Quadratic Function ARCHERY Ben shoots an arrow. The path of the arrow can be modeled by y = –16x 2 + 32x + 4, where y represents the height in feet of the arrow x seconds after it is shot into the air. A. Graph the height of the arrow. B. At what height was the arrow shot? C. What is the maximum height of the arrow?

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Example 6 Ellie hit a tennis ball into the air. The path of the ball can be modeled by y = –x 2 + 8x + 2, where y represents the height in feet of the ball x seconds after it is hit into the air. B. At what height was the arrow shot? C. What is the maximum height of the arrow? A. Graph the path of the ball.

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