1. Sec 2.4 Transformation of Graphs

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = –x 2 is the reflection of the graph of y = x 2 in the x-axis. Example: The graph of y = x 2 + 3 is the graph of y = x 2 shifted upward three units. This is a vertical shift. x y -4 4 4 -8 8 y = –x 2 y = x 2 + 3 y = x 2 Example: Shift, Reflection

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 f (x) f (x) + c +c+c f (x) – c -c If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units. Vertical Shifts If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units. x y Vertical Shifts

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 h(x) = |x| – 4 Example 1: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4. f (x) = |x| x y -4 4 4 8 g(x) = |x| + 3 Example: Vertical Shifts

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Graphing Utility: Vertical Shifts Ex 2. Match the functions with the correct graph. –55 4 –4

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 x y y = f (x)y = f (x – c) +c+c y = f (x + c) -c-c If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units. Horizontal Shifts If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units. Horizontal Shifts

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 f (x) = x 3 h(x) = (x + 4) 3 Ex 3: Use the graph of f (x) = x 3 to graph g (x) = (x – 2) 3 and h(x) = (x + 4) 3. x y -4 4 4 g(x) = (x – 2) 3 Example: Horizontal Shifts

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Graphing Utility: Horizontal Shifts Graphing Utility: Sketch the graphs given by –56 7 –1

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Ex 4: Describe the changes in the graph of the function to First make a vertical shift 4 units downward. Then a horizontal shift 5 units left. Example: Vertical and Horizontal Shifts

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 y = f (–x) y = f (x) y = –f (x) The graph of a function may be a reflection of the graph of a basic function. The graph of the function y = f ( – x) is the graph of y = f (x) reflected in the y-axis. The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x-axis. x y Reflection in the y-Axis and x-Axis.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 x y 4 4 y = x 2 y = – (x + 3) 2 Ex 5: Graph y = –(x + 3) 2 using the graph of y = x 2. First reflect the graph in the x-axis. Then shift the graph three units to the left. x y – 4 4 4 -4 y = – x 2 (–3, 0) Example: Reflections

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 HW #4 p190 1-4, 7, 11-13, 16, 23, 24, 27

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Vertical Stretching and Shrinking If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c. If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) shrunk vertically by c. Ex 1: y = 2x 2 is the graph of y = x 2 stretched vertically by 2. – 4– 4 x y 4 4 y = x 2 is the graph of y = x 2 shrunk vertically by. y = 2x 2 Vertical Stretching and Shrinking

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 - 4- 4 x y 4 4 y = |x| y = |2x| Horizontal Stretching and Shrinking If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by c. If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by c. Ex 2: y = |2x| is the graph of y = |x| shrunk horizontally by 2. is the graph of y = |x| stretched horizontally by. Example: Vertical Stretch and Shrink

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Graphing Utility: Vertical Stretch and Shrink Ex 3: Match the function to its graph. –5 5 5

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Ex 4: Explain how the graph of is obtained from y = x 3 Example: Multiple Transformations

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 HW #5 p190 5-10, 15, 17, 18, 25, 26, 28, 30, 32