Transformations of Graphs

1. 3.5 Transformations of Graphs Graph functions using vertical and horizontal translations Graph functions using stretching and shrinking Graph functions using reflections Combine transformations Model data with transformations

2. Parent Functions We will use these two graphs to demonstrate shifts, or translations, in the xy-plane. These are typically called “parent functions”.

3. Vertical Shift Examples A graph is shifted up or down. The shape of the graph is not changed—only its position.

4. Horizontal Shift Examples A graph is shifted right or left. The shape of the graph is not changed—only its position.

5. Combining Horizontal and Vertical Shifts Shifts can be combined to translate a graph of y = f(x) both vertically and horizontally. Shift the graph of y = x2 to the left 3 units and downward 2 units. y = x2y = (x + 3)2y = (x + 3)2 2

6. Combining Horizontal and Vertical Shifts Shifts can be combined to translate a graph of y = f(x) both vertically and horizontally. Shift the graph of y = |x| to the right 2 units and downward 4 units. y = |x|y = |x – 2|y = |x – 2|  4

7. Vertical Stretching and Shrinking If c > 1, the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) If 0 < c < 1 the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x).

8. Horizontal Stretching and Shrinking • If c > 1, the graph of y= f(cx) is a horizontal shrinking of the graph of y = f(x) • if 0 < c < 1 the graph of y = f(cx) is a horizontal stretching of the graph of y = f(x)

9. Example (vertical stretching) Use the graph of y = f(x) to sketch the graph of y = 3f(x) If c > 1, the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) If 0 < c < 1 the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x).

10. Example (horizontal stretching) Use the graph of y = f(x) to sketch the graph of • If c > 1, the graph of y= f(cx) is a horizontal shrinking of the graph of y = f(x) • if 0 < c < 1 the graph of y = f(cx) is a horizontal stretching of the graph of y = f(x)

11. Reflection of Graphs Across the x-Axis and y-Axis The graph of y = –f(x) is a reflection of the graph of y = f(x) across the x-axis. The graph of y = f(–x) is a reflection of the graph of y = f(x) across the y-axis.

12. Reflection Across X-Axis Make a table of values for y = –f(x) by negatingeach y-value in the table for f(x) Use the graph of y = f(x) to sketch the graph of y = –f(x)

13. Reflection Across Y-Axis Make a table of values for y = f(–x) by negatingeach x-value in the table for f(x) Use the graph of y = f(x) to sketch the graph of y = f(-x)

14. Combining Transformations The graph of y = 2(x – 1)2 + 3 can be obtained by performing four transformations on the graph of y = x2. 1. Shift the graph 1 unit right: y = (x – 1)2 2. Vertically stretch the graph by factor of 2: y = 2(x – 1)2 3. Reflect the graph across the x-axis: y = 2(x – 1)2 4. Shift the graph upward 3 units: y = 2(x – 1)2 + 3

15. Example (combined transformation) Describe how the graph of each equation can be obtained by transforming the graph of the parent equation Vertically shrink the graph by factor of 1/2 then reflect it across the x-axis.

16. Example (combined transformation graphs) Step 1. right 1 Step 2. stretch by 3 Step 3. up 2 The graph of y = –f(x) is a reflection of the graph of y = f(x) across the x-axis. The graph of y = f(–x) is a reflection of the graph of y = f(x) across the y-axis. If c > 1, the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) If 0 < c < 1 the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x).

17. Example (combined transformation graphs) Step 1. shrink by half Step 2. reflect Step 3. up 1 The graph of y = –f(x) is a reflection of the graph of y = f(x) across the x-axis. The graph of y = f(–x) is a reflection of the graph of y = f(x) across the y-axis. If c > 1, the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) If 0 < c < 1 the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x).

18. Example (combined transformation graphs) Step 1. right 3 Step 2. stretch by 2 Step 3. up 1 The graph of y = –f(x) is a reflection of the graph of y = f(x) across the x-axis. The graph of y = f(–x) is a reflection of the graph of y = f(x) across the y-axis. If c > 1, the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) If 0 < c < 1 the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x).

19. Key Ideas for this section: Transformation Animation • How do we do vertical and horizontal shifts of graphs? • How do we do vertical stretching and shrinking of graphs • How do we do horizontal stretching and shrinking of graphs • How do we do a reflections of graphs across the x-Axis or y-Axis • How do we combine transformations? • What do transformations tell us in the real world?

20. Example (combined transformation) Describe how the graph of each equation can be obtained by transforming the graph of the parent equation Reflect graph across the y-axis. Then shift left 2 units. Then shift down 1 unit.

21. Example(transforming backwards) Find an equation that shifts the graph of to the left 8 units and upwards 4 units. To shift the graph left 8 units, replace x with (x + 8) in the formula for f(x). To shift the graph upward 4 units, add 4 to the formula for f(x).