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Holt Algebra Transforming Quadratic Functions Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward. 1. y = x y = 2x 2 3. y = –0.5x 2 – 4 x = 0; (0, 3); opens upward x = 0; (0, 0); opens upward x = 0; (0, –4); opens downward

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Holt Algebra Transforming Quadratic Functions Students will be able to: Graph and transform quadratic functions. Learning Target

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Holt Algebra Transforming Quadratic Functions You saw in Lesson 5-9 that the graphs of all linear functions are transformations of the linear parent function y = x. Remember!

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Holt Algebra Transforming Quadratic Functions The quadratic parent function is f(x) = x 2. The graph of all other quadratic functions are transformations of the graph of f(x) = x 2. For the parent function f(x) = x 2 : The axis of symmetry is x = 0, or the y-axis. The vertex is (0, 0) The function has only one zero, 0.

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Holt Algebra Transforming Quadratic Functions

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Holt Algebra Transforming Quadratic Functions The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

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Holt Algebra Transforming Quadratic Functions Order the functions from narrowest graph to widest. f(x) = 3x 2, g(x) = 0.5x 2 Find |A| for each function. |3| = 3|0.05| = 0.05 f(x) = 3x 2 g(x) = 0.5x 2 The function with the narrowest graph has the greatest |A|.

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Holt Algebra Transforming Quadratic Functions Order the functions from narrowest graph to widest. f(x) = x 2, g(x) = x 2, h(x) = –2x 2 |1| = 1 |–2| = 2 The function with the narrowest graph has the greatest |A|. f(x) = x 2 h(x) = –2x 2 g(x) = x 2

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Holt Algebra Transforming Quadratic Functions Order the functions from narrowest graph to widest. f(x) = –x 2, g(x) = x 2 |–1| = 1 The function with the narrowest graph has the greatest |A|. f(x) = –x 2 g(x) = x 2

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Holt Algebra Transforming Quadratic Functions

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Holt Algebra Transforming Quadratic Functions The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-intercept but also a vertical translation of the graph of f(x) = ax 2 up or down the y-axis.

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Holt Algebra Transforming Quadratic Functions

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Holt Algebra Transforming Quadratic Functions When comparing graphs, it is helpful to draw them on the same coordinate plane. Helpful Hint

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Holt Algebra Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2. The graph of g(x) = x is wider than the graph of f(x) = x 2. g(x) = x The graph of g(x) = x opens downward.

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Holt Algebra Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2 g(x) = 3x 2

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Holt Algebra Transforming Quadratic Functions Compare the graph of each the graph of f(x) = x 2. g(x) = –x 2 – 4

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Holt Algebra Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2. g(x) = 3x 2 + 9

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Holt Algebra Transforming Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2. g(x) = x 2 + 2

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Holt Algebra Transforming Quadratic Functions The quadratic function h(t) = –16t 2 + c can be used to approximate the height h in feet above the ground of a falling object t seconds after it is dropped from a height of c feet. This model is used only to approximate the height of falling objects because it does not account for air resistance, wind, and other real-world factors.

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Holt Algebra Transforming Quadratic Functions Two identical softballs are dropped. The first is dropped from a height of 400 feet and the second is dropped from a height of 324 feet. a. Write the two height functions and compare their graphs. h 1 (t) = –16t Dropped from 400 feet. h 2 (t) = –16t Dropped from 324 feet.

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Holt Algebra Transforming Quadratic Functions The graph of h 2 is a vertical translation of the graph of h 1. Since the softball in h 1 is dropped from 76 feet higher than the one in h 2, the y- intercept of h 1 is 76 units higher. b. Use the graphs to tell when each softball reaches the ground.

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Holt Algebra Transforming Quadratic Functions Remember that the graphs show here represent the height of the objects over time, not the paths of the objects. Caution!

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