1. Shifting Graphs Digital Lesson

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: 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 Graphing Utility: Sketch the graphs given by –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 Example: 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 -4 y 4 x x y 4 Example: Graph the function using the graph of. First make a vertical shift 4 units downward. Then a horizontal shift 5 units left. (0, 0) (4, 2) (0, – 4) (4, –2) (– 5, –4) Example: Vertical and Horizontal Shifts (–1, –2)

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 Example: 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 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. Example: 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

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 - 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. Example: 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

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Graphing Utility: Vertical Stretch and Shrink Graphing Utility: Sketch the graphs given by –5 5 5

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 - 4- 4 4 4 8 x y Example: Graph using the graph of y = x 3. Step 4: - 4- 4 4 4 8 x y Step 1: y = x 3 Step 2: y = (x + 1) 3 Step 3: Example: Multiple Transformations Graph y = x 3 and do one transformation at a time.