1. College Algebra Chapter 2 Functions and Graphs
Section 2.6
Transformations of Graphs
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2. Concepts Recognize Basic Functions
Apply Vertical and Horizontal Translations (Shifts)
Apply Vertical and Horizontal Shrinking and Stretching
Apply Reflections Across the x- and y-Axes
Summarize Transformations of Graphs

3. Concept 1 Recognize Basic Functions

4. Recognize Basic Functions (1 of 4)
Linear function f(x) = mx + b
Constant function f(x) = b
Identity function
f(x) = x

5. Recognize Basic Functions (2 of 4)
Quadratic function
Cube function

6. Recognize Basic Functions (3 of 4)
Square root function
Cube root function

7. Recognize Basic Functions (4 of 4)
Absolute value function F(x) = |x|
Reciprocal function

8. Concept 2 Apply Vertical and Horizontal Translations (Shifts)

9. Apply Vertical and Horizontal Translations (Shifts) (1 of 2)
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.

10. Apply Vertical and Horizontal Translations (Shifts) (2 of 2)
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.

11. Example 1 (1 of 2)
Graph the functions.

12. Example 1 (2 of 2)
g(x) 2 up

13. Example 2 (1 of 2)
Graph the functions.

14. Example 2 (2 of 2)
h(x) 2 down
K(x) 2 right

15. Skill Practice 1
Use translations to graph the given function.

16. Skill Practice 2
Graph the function defined by g(x) = |x+2|.

17. Example 3 Graph the function.
Horizontal shift: 1 left Vertical shift: 4 down

18. Example 4
Graph the function. P(x) = |x - 2| + 3 Horizontal shift: 2 right Vertical shift: 3 up

19. Skill Practice 3
Use translations to graph the function defined by

20. Concept 3 Apply Vertical and Horizontal Shrinking and Stretching

21. Apply Vertical and Horizontal Shrinking and Stretching (1 of 2)
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.

22. Apply Vertical and Horizontal Shrinking and Stretching (2 of 2)
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 horizontal 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 horizontal by a factor of a.

23. Example 5 (1 of 2)
Graph the functions.

24. Example 5 (2 of 2)

25. Skill Practice 4
Graph the functions.

26. Example 6 (1 of 2)
Graph the functions.

27. Example 6 (2 of 2)

28. Example 7
Use the graph of f(x) to graph y=f(4x)

29. Skill Practice 5
The graph of y=f(x) is shown. Graph.

30. Concept 4 Apply Reflections Across the x- and y-Axes

31. 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.

32. Example 8 (1 of 2)
The graph of y=f(x) is given Graph y=-f(x) and y=f(-x)

33. Example 8 (2 of 2)

34. Skill Practice 6 The graph of y=f(x) is given. Graph y=-f(x).

35. Concept 5 Summarize Transformations of Graphs

36. Summarize of Transformations of Graphs (1 of 6)
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
Shift upward c units
Replace (x, y) by (x, y + c)
y = f(x) – c
Shift downward c units
Replace (x, y) by (x, y – c)

37. Summarize of Transformations of Graphs (2 of 6)
Effect on the Graph of f
Changes to Points on f
Horizontal translation (shift)
y = f(x-h)
Shift right h units
Replace (x, y) by (x + h, y).
y = f(x + h)
Shift left h units
Replace (x, y) by (x – h, y).

38. Summarize of Transformations of Graphs (3 of 6)
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).

39. Summarize of Transformations of Graphs (4 of 6)

40. Summarize of Transformations of Graphs (5 of 6)
Effect on the Graph of f
Changes to Points on f
Reflection
y = -f(x)
Reflection across the x-axis
Replace (x, y) by (x, -y).
y = f(-x)
Reflection across the y-axis
Replace (x, y) by (-x, y).

41. Summarize of Transformations of Graphs (6 of 6)
To graph a function requiring multiple transformations, use the following order.
Horizontal translation (shift)
Horizontal and vertical stretch and shrink
Reflections across x- or y-axis
Vertical translation (shift)

42. Example 9 (1 of 3)
Graph the function defined by f(x) =2|x + 1|-3 Parent function: f(x)=|x|

43. Example 9 (2 of 3)
f(x) = 2|x + 1|-3 Shift the graph to the left 1 unit
f(x) = 2|x + 1|-3 Apply a vertical stretch
(multiply the y-values by 2)
f(x) = 2|x + 1|-3 Shift the graph downward 3 units

44. Example 9 (3 of 3)

45. Skill Practice 7
Use transformation to graph the function defined by m(x) = -3|x - 2| -4.

46. Example 10 (1 of 3)
Graph the function defined by
Parent function:

47. Example 10 (2 of 3)
- → vertical reflect. ½ → vertical shrink -x → horizontal reflect 3 → Horizontal shift 3 right

48. Example 10 (3 of 3)

49. Skill Practice 8
Use transformations to graph the function defined by