1. College Algebra Chapter 2 Functions and Graphs
Section 2.6
Transformations of Graphs

2. 1. Recognize Basic Functions
2. Apply Vertical and Horizontal Translations (Shifts)
3. Apply Vertical and Horizontal Shrinking and Stretching
4. Apply Reflections Across the x- and y-Axes
5. Summarize Transformations of Graphs

3. Recognize Basic Functions
Linear function
Constant function
Identity function

4. Recognize Basic Functions
Quadratic function
Cube function

5. Recognize Basic Functions
Square root function
Cube root function

6. Recognize Basic Functions
Absolute value function
Reciprocal function

7. 1. Recognize Basic Functions
2. Apply Vertical and Horizontal Translations (Shifts)
3. Apply Vertical and Horizontal Shrinking and Stretching
4. Apply Reflections Across the x- and y-Axes
5. Summarize Transformations of Graphs

8. Apply Vertical and Horizontal Translations (Shifts)
Consider a function defined by y = f(x).
Let c and h represent positive real numbers.
Vertical shift:
The graph of y = f(x) + c is the graph of y = f(x) shifted c units upward.
The graph of y = f(x) – c is the graph of y = f(x) shifted c units downward.

9. Apply Vertical and Horizontal Translations (Shifts)
Consider a function defined by y = f(x).
Let c and h represent positive real numbers.
Horizontal shift:
The graph of y = f(x – h) is the graph of y = f(x) shifted h units to the right.
The graph of y = f(x + h) is the graph of y = f(x) shifted h units to the left.

10. Example 1:
Graph the functions.
Parent
Families x 1 2 3 4

11. Example 1 continued:

12. Example 2:
Graph the functions.
Parent
Families x 1 2 3 4

13. Example 2 continued:

14. Example 3:
Graph the function.
Horizontal shift: ___________
Vertical shift: _____________

15. Example 4:
Graph the function.
Horizontal shift: ___________
Vertical shift: _____________

16. 1. Recognize Basic Functions
2. Apply Vertical and Horizontal Translations (Shifts)
3. Apply Vertical and Horizontal Shrinking and Stretching
4. Apply Reflections Across the x- and y-Axes
5. Summarize Transformations of Graphs

17. Apply Vertical and Horizontal Shrinking and Stretching
Consider a function defined by y = f(x).
Let a represent a positive real number.
Vertical shrink/stretch:
If a > 1 , then the graph of y = a f(x) is the graph of y = f(x) stretched vertically by a factor of a.
If 0 < a < 1 , then the graph of y = a f(x) is the graph of y = f(x) shrunk vertically by a factor of a.

18. Example 5:
Graph the functions. x 1 2 3 4 x 1 2 3 4

19. Example 5 continued:

20. Apply Vertical and Horizontal Shrinking and Stretching
Consider a function defined by y = f(x).
Let a represent a positive real number.
Horizontal shrink/stretch:
If a > 1 , then the graph of y = f(a  x) is the graph of
y = f(x) shrunk horizontally by a factor of a.
If 0 < a < 1 , then the graph of y = f(a  x) is the graph of y = f(x) stretched horizontally by a factor of a.

21. Example 6:
Graph the functions. x 1 2 3 4 x 1 2 3 4

22. Example 6 continued:

23. Example 7:

24. 1. Recognize Basic Functions
2. Apply Vertical and Horizontal Translations (Shifts)
3. Apply Vertical and Horizontal Shrinking and Stretching
4. Apply Reflections Across the x- and y-Axes
5. Summarize Transformations of Graphs

25. Apply Reflections Across the x- and y-Axes
Consider a function defined by y = f(x).
Reflection across the x-axis:
The graph of y = – f(x) is the graph of y = f(x) reflected across the x-axis.
Reflection across the y-axis:
The graph of y = f(– x) is the graph of y = f(x) reflected across the y-axis.

26. Example 8:

27. 1. Recognize Basic Functions
2. Apply Vertical and Horizontal Translations (Shifts)
3. Apply Vertical and Horizontal Shrinking and Stretching
4. Apply Reflections Across the x- and y-Axes
5. Summarize Transformations of Graphs

28. Summarize of Transformations of Graphs
Transformations of Functions
Consider a function defined by y = f(x). If c, h, and a represent positive real numbers, then the graphs of the following functions are related to y = f(x) as follows.
Transformation
Effect on the Graph of f
Changes to Points on f
Vertical translation (shift)
y = f(x) + c
y = f(x) – c
Shift upward c units
Shift downward c units
Replace (x, y) by (x, y + c)
Replace (x, y) by (x, y – c)

29. Summarize of Transformations of Graphs
Transformations of Functions
Consider a function defined by y = f(x). If c, h, and a represent positive real numbers, then the graphs of the following functions are related to y = f(x) as follows.
Transformation
Effect on the Graph of f
Changes to Points on f
Horizontal translation (shift)
y = f(x – h)
y = f(x + h)
Shift right h units
Shift left h units
Replace (x, y) by (x + h, y).
Replace (x, y) by (x – h, y).

30. Summarize of Transformations of Graphs
Transformations of Functions
Consider a function defined by y = f(x). If c, h, and a represent positive real numbers, then the graphs of the following functions are related to y = f(x) as follows.
Transformation
Effect on the
Graph of f
Changes to Points on f
Vertical stretch/shrink
y = a[f(x)]
Vertical stretch (if a > 1)
Vertical shrink (if 0 < a < 1)
Graph is stretched/ shrunk vertically by a factor of a.
Replace (x, y) by (x, ay).

31. Summarize of Transformations of Graphs
Transformations of Functions
Consider a function defined by y = f(x). If c, h, and a represent positive real numbers, then the graphs of the following functions are related to y = f(x) as follows.
Transformation
Effect on the
Graph of f
Changes to Points on f
Horizontal stretch/shrink
y = f(ax)
H. shrink (if a > 1)
H. stretch (if 0 < a < 1)
Graph is stretched/shrunk horizontally by a factor of .
Replace (x, y) by

32. Summarize of Transformations of Graphs
Transformations of Functions
Consider a function defined by y = f(x). If c, h, and a represent positive real numbers, then the graphs of the following functions are related to y = f(x) as follows.
Transformation
Effect on the
Graph of f
Changes to Points on f
Reflection
y = –f(x)
y = f(–x)
Reflection across the x-axis
Reflection across the y-axis
Replace (x, y) by (x, –y).
Replace (x, y) by (–x, y).

33. Summarize of Transformations of Graphs
To graph a function requiring multiple transformations, use the following order.
1. Horizontal translation (shift)
2. Horizontal and vertical stretch and shrink
3. Reflections across x- or y-axis
4. Vertical translation (shift)

34. .
Example 9:
Parent function:

35. Shift the graph to the left 1 unit.
Example 9 continued:
Shift the graph to the left 1 unit
Apply a vertical stretch
(multiply the y-values by 2)
Shift the graph downward 3 units

36. .
Example 10:
Parent function:

37. .
Example 10 continued: