Unit 1.6
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This document discusses various types of graphical transformations, including translations, reflections, stretches, and shrinks. It provides examples of applying each type of transformation to graphs and finding the equations that result. Key points covered include translating graphs vertically or horizontally by adding or subtracting values, reflecting graphs across the x-axis or y-axis, stretching or shrinking graphs by multiplying values, and combining multiple transformations in sequence to transform one graph into another.
Unit 1.6
- 1. 1.6 Graphical Transformations Copyright © 2011 Pearson, Inc.
- 2. What you’ll learn about Transformations Vertical and Horizontal Translations Reflections Across Axes Vertical and Horizontal Stretches and Shrinks Combining Transformations … and why Studying transformations will help you to understand the relationships between graphs that have similarities but are not the same. Copyright © 2011 Pearson, Inc. Slide 1.6 - 2
- 3. Transformations In this section we relate graphs using transformations, which are functions that map real numbers to real numbers. Rigid transformations, which leave the size and shape of a graph unchanged, include horizontal translations, vertical translations, reflections, or any combination of these. Nonrigid transformations, which generally distort the shape of a graph, include horizontal or vertical stretches and shrinks. Copyright © 2011 Pearson, Inc. Slide 1.6 - 3
- 4. Vertical and Horizontal Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x). Horizontal Translations y = f(x – c) a translation to the right by c units y = f(x + c) a translation to the left by c units Vertical Translations y = f(x) + c a translation up by c units y = f(x) – c a translation down by c units Copyright © 2011 Pearson, Inc. Slide 1.6 - 4
- 5. Example Vertical Translations Describe how the graph of f (x) x can be transformed to the graph of y x 4. Copyright © 2011 Pearson, Inc. Slide 1.6 - 5
- 6. Solution Describe how the graph of f (x) x can be transformed to the graph of y x 4. It is a translation down by 4 units. Copyright © 2011 Pearson, Inc. Slide 1.6 - 6
- 7. Example Finding Equations for Translations Each view shows the graph of y1 x3 and a vertical or horizontal translation y2 . Write an equation for y2 . Copyright © 2011 Pearson, Inc. Slide 1.6 - 7
- 8. Solution Each view shows the graph of y1 x3 and a vertical or horizontal translation y2 . Write an equation for y2 . (a) y2 x3 3 (b) y2 x 23 (c) y2 x 33 Copyright © 2011 Pearson, Inc. Slide 1.6 - 8
- 9. Reflections The following transformations result in reflections of the graph of y = f(x): Across the x-axis y = –f(x) Across the y-axis y = f(–x) Copyright © 2011 Pearson, Inc. Slide 1.6 - 9
- 10. Graphing Absolute Value Compositions Given the graph of y = f(x), the graph y = |f(x)| can be obtained by reflecting the portion of the graph below the x-axis across the x-axis, leaving the portion above the x-axis unchanged; the graph of y = f(|x|) can be obtained by replacing the portion of the graph to the left of the y-axis by a reflection of the portion to the right of the y-axis across the y-axis, leaving the portion to the right of the y-axis unchanged. (The result will show even symmetry.) Copyright © 2011 Pearson, Inc. Slide 1.6 - 10
- 11. Stretches and Shrinks Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph of y f (x): Horizontal Stretches or Shrinks y f x c a stretch by a factor of c if c 1 a shrink by a factor of c if c 1 Vertical Stretches or Shrinks y c f (x) a stretch by a factor of c if c 1 a shrink by a factor of c if c 1 Copyright © 2011 Pearson, Inc. Slide 1.6 - 11
- 12. Example Finding Equations for Stretches and Shrinks Let C1 be the curve defined by y1 x3 3. Find equations for the following non-rigid transformations of C1 : (a) C2 is a vertical stretch of C1 by a factor of 4. (b) C3 is a horizontal shrink of C1 by a factor of 1/3. Copyright © 2011 Pearson, Inc. Slide 1.6 - 12
- 13. Solution Let C1 be the curve defined by y1 x3 3. Find equations for the following non-rigid transformations of C1 : (a) C2 is a vertical stretch of C1 by a factor of 4. (b) C3 is a horizontal shrink of C1 by a factor of 1/3. (a) y2 4 f (x) 4(x3 3) 4x3 12 (b) y3 f x 1 / 3 f (3x) (3x)3 3 27x3 3 Copyright © 2011 Pearson, Inc. Slide 1.6 - 13
- 14. Example Combining Transformations in Order The graph of y x2 undergoes the following transformations, in order. Find the equation of the graph that results. a horizontal shift 5 units to the left a vertical stretch by a factor of 3 a vertical translation 4 units up Copyright © 2011 Pearson, Inc. Slide 1.6 - 14
- 15. Solution The graph of y x2 undergoes the following transformations, in order. Find the equation of the graph that results. a horizontal shift 5 units to the left a vertical stretch by a factor of 3 a vertical translation 4 units up x2 x 52 3x 52 3x 52 4 Expanding the final expression: y 3x2 30x 79 Copyright © 2011 Pearson, Inc. Slide 1.6 - 15
- 16. Example Combining Transformations in Order Describe how to transform the graph of y f x shown to the graph of y f x 2 4. Copyright © 2011 Pearson, Inc. Slide 1.6 - 16
- 17. Solution Describe how to transform the graph of y f x shown to the graph of y f x 2 4. Copyright © 2011 Pearson, Inc. Slide 1.6 - 17
- 18. Solution (continued) Describe how to transform the graph of y f x shown to the graph of y f x 2 4. Copyright © 2011 Pearson, Inc. Slide 1.6 - 18
- 19. Quick Review Write the expression as a binomial squared. 1. x2 4x 4 2. x2 2x 1 3. 4x2 36x 81 Perform the indicated operations and simplify. 4. (x 1)2 (x 1) 2 5. (x 1)3 (x 1) 2 Copyright © 2011 Pearson, Inc. Slide 1.6 - 19
- 20. Quick Review Solutions Write the expression as a binomial squared. 1. x2 4x 4 (x 2)2 2. x2 2x 1 (x 1)2 3. 4x2 36x 81 (2x 9)2 Perform the indicated operations and simplify. 4. (x 1)2 (x 1) 2 x2 x 2 5. (x 1)3 (x 1) 2 x3 3x2 4x Copyright © 2011 Pearson, Inc. Slide 1.6 - 20