1. Absolute Value Functions and Graphs Lesson 2-7
Algebra 2
Absolute Value Functions and Graphs
Lesson 2-7

2. Goals Goal Rubric To graph absolute value functions.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Absolute Value Function
Axis of Symmetry
Vertex

4. Essential Question Big Idea: Function
What is an absolute value function?

5. Definitions
Absolute-Value Function - is a function whose rule contains an absolute-value expression.
The graph of the parent absolute-value function f(x) = |x| has a V shape and is symmetric about a vertical line called the axis of symmetry.
The graph has either a single maximum point or a single minimum point, called the vertex. The parent absolute-value function has a vertex at (0, 0).

6. Absolute-Value Function

7. Absolute-Value Function
The absolute-value parent function is composed of two linear pieces, one with a slope of –1 and one with a slope of 1. You can transform absolute-value functions.

8. The general forms for translations are
Vertical:
g(x) = f(x) + k
Horizontal:
g(x) = f(x – h)
Remember!

9. Example:
Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).
5 units down
f(x) = |x|
g(x) = f(x) + k
g(x) = |x| – 5
Substitute.
The graph of g(x) = |x| – 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5).

10. Example: continued
The graph of g(x) = |x|– 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5).
f(x)
g(x)

11. Example:
Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).
1 unit left
f(x) = |x|
g(x) = f(x – h )
g(x) = |x – (–1)| = |x + 1|
Substitute.

12. Example: continued
The graph of g(x) = |x + 1| is the graph of f(x) = |x| after a horizontal shift of 1 unit left. The vertex of g(x) is (–1, 0).
f(x)
g(x)

13. Your Turn:
Let g(x) be the indicated transformation of f(x) = |x|. Write the rule for g(x) and graph the function.
4 units down
f(x) = |x|
g(x) = f(x) + k
g(x) = |x| – 4
Substitute.

14. Your Turn: continued
The graph of g(x) = |x| – 4 is the graph of f(x) = |x| after a vertical shift of 4 units down. The vertex of g(x) is (0, –4).
f(x)
g(x)

15. Your Turn:
Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).
2 units right
f(x) = |x|
g(x) = f(x – h)
g(x) = |x – 2| = |x – 2|
Substitute.

16. Your Turn: continued
The graph of g(x) = |x – 2| is the graph of f(x) = |x| after a horizontal shift of 2 units right. The vertex of g(x) is (2, 0).
f(x)
g(x)

17. Absolute-Value Function Vertex
Because the entire graph moves when shifted, the shift from f(x) = |x| determines the vertex of an absolute-value graph.

18. Example:
Translate f(x) = |x| so that the vertex is at (–1, –3). Then graph.
g(x) = |x – h| + k
g(x) = |x – (–1)| + (–3)
Substitute.
g(x) = |x + 1| – 3

19. Example: continued
The graph of g(x) = |x + 1| – 3 is the graph of f(x) = |x| after a vertical shift down 3 units and a horizontal shift left 1 unit.
f(x)
The graph confirms that the vertex is (–1, –3).
g(x)

20. Your Turn:
Translate f(x) = |x| so that the vertex is at (4, –2). Then graph.
g(x) = |x – h| + k
g(x) = |x – 4| + (–2)
Substitute.
g(x) = |x – 4| – 2

21. Your Turn: continued
The graph of g(x) = |x – 4| – 2 is the graph of f(x) = |x| after a vertical down shift 2 units and a horizontal shift right 4 units.
g(x)
f(x)
The graph confirms that the vertex is (4, –2).

22. Absolute-value functions can also be stretched, compressed, and reflected.
Reflection across x-axis: g(x) = –f(x)
Reflection across y-axis: g(x) = f(–x)
Remember!
Vertical stretch and compression : g(x) = af(x)
Remember!

23. Example: Perform the transformation. Then graph.
Reflect the graph. f(x) =|x – 2| + 3 across the y-axis.
g(x) = f(–x)
Take the opposite of the input value.
g(x) = |(–x) – 2| + 3

24. Example: continued
The vertex of the graph g(x) = |–x – 2| + 3 is (–2, 3). f g

25. Example:
Stretch the graph. f(x) = |x| – 1 vertically by a factor of 2.
g(x) = af(x)
g(x) = 2(|x| – 1)
Multiply the entire function by 2.
g(x) = 2|x| – 2

26. Example: continued
The graph of g(x) = 2|x| – 2 is the graph of f(x) = |x| – 1 after a vertical stretch by a factor of 2. The vertex of g is at (0, –2).
g(x)
f(x)

27. Your Turn: Perform the transformation. Then graph.
Reflect the graph. f(x) = –|x – 4| + 3 across the y-axis.
g(x) = f(–x)
Take the opposite of the input value.
g(x) = –|(–x) – 4| + 3
g(x) = –|–x – 4| + 3

28. Your Turn: continued
The vertex of the graph g(x) = –|–x – 4| + 3 is (–4, 3). g f

29. Your Turn:
Compress the graph of f(x) = |x| + 1 vertically by a factor of .
g(x) = a(|x| + 1)
g(x) = (|x| + 1)
Multiply the entire function by .
g(x) = ( |x| + )
Simplify.

30. Your Turn: continued
The graph of g(x) = |x| + is the graph of g(x) = |x| + 1 after a vertical compression by a factor of The vertex of g is at ( 0, ).
f(x)
g(x)

31. Absolute-Value Function Transformations

32. Your Turn: How does this transform the graph? Then graph.
Reflect graph across the x axis:

33. Your Turn: How does this transform the graph? Then graph.
Vertically stretches by a factor of 2:

34. Your Turn: How does this transform the graph? Then graph.
Vertically compression by a factor of ½ :

35. Your Turn: How does this transform the graph? Then Graph.
Moves it horizontally one unit to the right:

36. Your Turn: How does this transform the graph? Then graph.
Translates it vertically down 4 units:

37. Your Turn: Perform the transformation. Then graph.
Translate f(x) = |x| by 3 units right. f g
g(x)=|x – 3|

38. Your Turn: Perform each transformation. Then graph.
Translate f(x) = |x| so the vertex is at (2, –1) Then graph. f g
g(x)=|x – 2| – 1

39. Your Turn:
Graph
y = |x+1| + 2 2 1

40. y = 2 |x - 3| + 1 Your Turn: Graph 1 3 stretch stretch Slope: 2

41. Your Turn: How does this transform the graph? Then graph.
Horizontally translates 1 unit left
Vertically translates 3 units down
Vertically stretches by 2:

42. Absolute-Value Function Transformation Summary:
f(x) = ± a|x-b|+c
Vertex (b,c)
b horizontal translation
c vertical translation
Direction:
+ opens up
- Opens down
Slope of sides

43. Essential Question Big Idea: Function
What is an absolute value function?
An absolute-value function gives the distance from the line y = 0 for each value of f(x). Write the equation of the function in the general form y = a|x – h| + k to identify the transformations of the parent function f(x) = |x|.

44. Assignment
Section 2-7, Pg 125 – 127; #1 – 7 all, 8 – 42 even.