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8.3 Notes Handout

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(page 453) Translations move points and graphs around the coordinate plane. Have you noticed that the image of the translation always looks like the original figure? Although the image of a translation moves, it doesn’t flip, turn, or change size. To get these changes, you need other types of transformations. Lesson 8.3 is titled Reflecting Points and Graphs. What do you think will happen to a figure when it is reflected?

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**1. Reflect the figure across the x-axis.**

y x’ y’ X Y Z X’ Y’ Z’ Describe the change in the ordered pairs of the pre-image of the figure to the image of the figure. Pre-image (x, y) and the image is (x, -y) The y-coordinate changed signs.

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**2. Reflect the figure across the y-axis.**

Z’ X Y Z Describe the change in the ordered pairs of the pre-image of the figure to the image of the figure. Pre-image (x, y) and the image is (-x, y) The x-coordinate changed signs, but zero stayed the same.

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**3. Reflect the figure across the x-axis and y-axis.**

Z X Y Z Describe the change in the ordered pairs of the pre-image of the figure to the image of the figure. Pre-image (x, y) and the image is (-x, -y) Both the x –coordinates and the y-coordinates changed signs, but zero stayed the same.

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**4. Reflect y = 2x across the x-axis.**

0 1 2 4 8 0 -1 -2 -4 -8 Describe the change in the ordered pairs of the pre-image of the figure to the image of the figure. The y-coordinate changed signs.

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**5. Reflect y = 2x across the y-axis.**

0 1 2 4 8 Describe the change in the ordered pairs of the pre-image of the figure to the image of the figure. The x-coordinate changed signs, but zero stayed the same.

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**A transformation that flips a figure to create a mirror image is called a reflection.**

A point is reflected across the x-axis, or vertically reflected, when you change the sign of its y-coordinate. A point is reflected across the y-axis, or horizontally reflected, when you change the sign of its x-coordinate. You saw both types of reflections in the investigation. Similar reflections result when you change the sign of x or y in a function. You can combine reflections with other transformations. Sometimes, different combinations will give the same result.

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**Some reflections will not change the graph. **

Does y = │– x│ have the same graph as y = │x│? Why or why not? Does y = ( – x)² have the same graph as y = x²? Why or why not? Does – y = │ x│ have the same graph as y = │x│? Why or why not? Does – y = x² have the same graph as y = x²? Why or why not? Yes the domain and range stay the same Yes the domain and range stay the same No, the range changes No, the range changes We will normally write – y = │ x│as y = –│ x│and – y = x² as y = – x².

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**Given the equation describe the transformation. **

6. f(x) = -(x – 2)² + 3 – y = │x + 4│+ 2 8. y – 3 = 4-x 9. 10. Reflect across the x-axis, translation right 2 units and up 3 units Reflect across the x-axis, translation left 4 units and up 2 units Reflect across the y-axis, translation up 3 units translation right 2 units Reflect across the x-axis, translation right 3 units and up 10 units

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**Reflection across the x-axis, Translation left 3 units **

Describe the transformation and then write a function for the image. Reflection across the x-axis, Translation left 3 units Vertex (3, 0) a = -1 y = – 1(x – 3)² Reflection across the x-axis, Translation right 7 units and down 3 units Vertex (-7, -3) a = -1 y = – 1(x + 7)² – 3

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Reflection across the x-axis, Translation right 4 units and down 2 Vertex (4, -2) a = -1 y = – 1|x – 4| – 2 Reflection across the x-axis, Translation right 3 units and up 6 Vertex (3, 6) a = -1 y = – 1|x – 3| + 6

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**Reflection across the x-axis, Translation right 4 units y = –1(3x – 4) **

Reflection across the x-axis, Translation right 4 units y = –1(3x – 4) Translation right 8 units and down 6 units y = f(x – 8) – 6

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**Reflection across the x-axis, Translation left 5 units **

Reflection across the x-axis, Translation left 5 units Reflection across the x-axis, Translation right 4 units and up 2 units

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