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Transformations of Functions Students will be able to draw graphs of functions after a transformation.

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FHS Functions 2 In this lesson we will be investigating transformations of functions. A transformation is a change in the position, size, or shape of a figure. A translation, or slide, is a transformation that moves each point in a figure the same distance in the same direction. Transformations

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FHS Functions 3 Translate the point (–3, 4) 5 units to the right. Give the coordinates of the translated point. Example 1 Translating (–3, 4) 5 units right results in the point (2, 4). (2, 4) 5 units right (-3, 4)

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FHS Functions 4 Translate the point (–3, 4) 2 units left and 3 units down. Give the coordinates of the translated point. Translating (–3, 4) 2 units left and 3 units down results in the point (–5, 1). (–3, 4) (–5, 1) 2 units 3 units Example 2

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Notice that when you translate left or right, the x-coordinate changes, and when you translate up or down, the y-coordinate changes. Translations Horizontal TranslationVertical Translation Translations

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FHS Functions 6 A reflection is a transformation that flips a figure across a line called the line of reflection. Each reflected point is the same distance from the line of reflection, but on the opposite side of the line. Reflection A reflection across the y-axis would look like this:

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FHS Functions 7 A reflection across the x-axis would look like this: Reflection

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FHS Functions 8 Reflection across x-axis Identify important points from the graph and make a table. xy–y–y –5–3–1(–3) = 3 –20– 1(0) = 0 0–2– 1(–2) = 2 20 – 1(0) = 0 5–3 – 1(–3) = 3 Multiply each y-coordinate by – 1. The entire graph flips across the x-axis. Example 3

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FHS Functions 9 Imagine grasping two points on the graph of a function that lie on opposite sides of the y-axis. If you pull the points away from the y-axis, you would create a horizontal stretch of the graph. If you push the points towards the y-axis, you would create a horizontal compression. Stretch/Compression

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FHS Functions 10 Similarily, if you pull points away from the x-axis, you would create a vertical stretch. If you push points toward the x-axis, you would create a vertical compression. Vertical stretches are similar to horizontal compressions, and vertical compressions are similar to horizontal stretches. Stretches and compressions are not congruent to the original graph. Stretch/Compression

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FHS Functions 11 Stretch/Compression

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FHS Functions 12 Use a table to perform a horizontal stretch of the function y = f(x) by a factor of 3. Graph the function and the transformation on the same coordinate plane. Multiply each x-coordinate by 3. Identify important points from the graph and make a table. 3x3xxy 3(–1) = –3–13 3(0) = 000 3(2) = 6 22 3(4) = 1242 Example 4

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FHS Functions 13 Example 5 Identify important points from the graph and make a table. Use a table to perform a vertical stretch of y = f(x) by a factor of 2. Graph the transformed function on the same coordinate plane as the original figure. xy2y2y –132(3) = 6 002(0) = 0 222(2) = 4 42 Multiply each y-coordinate by 2.

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