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Holt Algebra 1 9-4 Transforming Quadratic Functions Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function.

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Presentation on theme: "Holt Algebra 1 9-4 Transforming Quadratic Functions Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function."— Presentation transcript:

1 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

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

3 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!

4 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.

5 Holt Algebra Transforming Quadratic Functions

6 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.

7 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|.

8 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

9 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

10 Holt Algebra Transforming Quadratic Functions

11 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.

12 Holt Algebra Transforming Quadratic Functions

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

14 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.

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

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

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

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

19 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.

20 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.

21 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.

22 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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