1. Sullivan PreCalculus Section 2.5 Graphing Techniques: Transformations
Objectives
Graph Functions Using Horizontal and Vertical Shifts
Graph Functions Using Compressions and Stretches
Graph Functions Using Reflections about the x-Axis and y-Axis

2. Example: Use the graph of to obtain the
First, find points on the graph of f and g.

3. (2, 6)
(1, 3)
(2, 4)
(0, 2)
(1, 1)
If a real number c is added to the right side of a function y = f(x), the graph of the new function y = f(x) + c is the graph of f shifted vertically up (for c > 0).

4. Example: Use the graph of to obtain the
First, find points on the graph of f and g.

5. (2, 4)
(1, 1)
(2, 1)
(1, -2)
(0, -3)
If a real number c is subtracted from the right side of a function y = f(x), the graph of the new function y = f(x) - c is the graph of f shifted vertically down (for c > 0).

6. Example: Use the graph of to obtain the
First, find points on the graph of f and g.

7. (0, 4)
(2, 4)
(-1, 1)
(-3, 1)
(1, 1)
(-2, 0)
If a real number c is subtracted from the argument x of a function y = f(x), the graph of the new function y = f(x - c) is the graph of f shifted horizontally right (if c > 0) or left (if c < 0).

8. Example: Graph the function
(0, 0)
(2, 4)
(-3, 0)
(-1, 4)
(-3, -2)
(-1, 2)

9. Example: Use the graph of to obtain the
First, find points on the graph of f and g.

10. (2, 4)
(1, 2)
(2, 2)
(1, 1)
When the right side of a function y = f(x) is multiplied by a positive number k, the graph of the new function y = kf (x) is a vertically compressed (if 0 < k < 1) vertically stretched (if k > 1) version of the graph of y = f(x).

11. Example: Graph each of the following functions:Y2 Y1 Y3
When the argument of a function y = f (x) is multiplied by a positive number k, the graph of the new function y = f(kx) is a horizontally stretched (if 0 < k < 1) or horizontally compressed (if k > 1) version of the graph of y = f (x).

12. Example: Graph
(4, 2)
(4, 6)
(8, 6)

13. Example: Use the graph of to obtain the
First, find points on the graph of f and g.

14. (-3, 9)
(2, 4)
(2, -4)
(-3, -9)
When the right side of a function y = f(x) is multiplied by -1, the graph of the new function y = -f (x) is the reflection about the x-axis of the graph of the function y = f(x).

15. Example: Use the graph of to obtain the
First, find points on the graph of f and g.

16. (9, 3)
(-9, 3)
(-4, 2)
(4, 2)
(-1, 1)
(1, 1)
When the argument of a function y = f (x) is multiplied by -1, the graph of the new function y = f(-x) is the reflection about the y-axis of the graph of y = f (x).

17. Example: Graph
(1,1)
(4,2)
(-1,1)
(-4,2)
(-1,-1)
(-4,-2)