1. Absolute Value Functions
Algebra II
Chapter 02
A BowerPoint Presentation

2. The graph of y = |x|

3. The graph of y = |x|
When x is 3, what is y?

4. The graph of y = |x|
When x is 3, what is y?
When x is -3, what is y?

5. The graph of y = |x| When x is 3, what is y? When x is -3, what is y?
What point is the VERTEX of this function?

6. The graph of y = |x| When x is 3, what is y? When x is -3, what is y?
What point is the VERTEX of this function?
What is the slope of the right-side ray?

7. Let’s make a table of points
The graph of y = 2|x – 1| + 3
Let’s make a table of points

8. Let’s make a table of points
The graph of y = 2|x – 1| + 3
Let’s make a table of points
X Y

9. Let’s make a table of points
The graph of y = 2|x – 1| + 3
Let’s make a table of points
X Y -1 1 2 3
Find the corresponding y values

10. Let’s make a table of points
The graph of y = 2|x – 1| + 3
Let’s make a table of points
X Y -1 1 2 3 7 5 3
Do you notice anything?

11. Let’s make a graph using those points
The graph of y = 2|x – 1| + 3
Let’s make a graph using those points

12. The graph of y = 2|x – 1| + 3

13. The graph of y = 2|x – 1| + 3
What is the VERTEX of this graph?

14. The graph of y = 2|x – 1| + 3 What is the VERTEX of this graph?
What is the slope of the right-side ray?

15. The graph of y = 2|x – 1| + 3 What is the VERTEX of this graph?
What is the slope of the right-side ray?
Does this graph open UP or DOWN?

16. The graph of y = 2|x – 1| + 3 What is the VERTEX of this graph?
What is the slope of the right-side ray?
Does this graph open UP or DOWN?
Is this graph WIDER, NARROWER, or THE SAME as the graph of y = |x|?

17. What’s up w/absolute value functions
y = a | x – h | + k Do you see how this looks like y – y1 = m (x – x1) ? [Maybe not yet – let’s move y1…]

18. What’s up w/absolute value functions
y = a | x – h | + k Do you see how this looks like y – y1 = m (x – x1) ?

19. What’s up w/absolute value functions
y = a | x – h | + k Do you see how this looks like y= m (x – x1) + y1 ?

20. What’s up w/absolute value functions
y = a | x – h | + k Do you see how this looks like y= m (x – x1) + y1 ?

21. What’s up w/absolute value functions
y = a | x – h | + k The vertex of this graph will be the point (h, k)

22. What’s up w/absolute value functions
y = a | x – h | + k The slope of the right-side ray will be a

23. What’s up w/absolute value functions
y = a | x – h | + k The slope of the right-side ray will be a The slope of the left-side ray will be -a

24. What’s up w/absolute value functions
y = a | x – h | + k If a is POSITIVE If a is NEGATIVE Graph opens Graph opens UP DOWN

25. What’s up w/absolute value functions
y = a | x – h | + k If |a| > 1 If |a| = 1 If |a| < 1 Narrower Same width Wider than y =|x| than y =|x| than y =|x|

26. Graph this absolute value function: y = – |x + 2| – 3
Let’s graph!
Graph this absolute value function: y = – |x + 2| – 3

27. Graph this absolute value function: y = – |x + 2| – 3
Let’s graph!
Graph this absolute value function: y = – |x + 2| – 3
Will this graph open UP or DOWN?

28. Graph this absolute value function: y = – |x + 2| – 3
Let’s graph!
Graph this absolute value function: y = – |x + 2| – 3
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?

29. Graph this absolute value function: y = – |x + 2| – 3
Let’s graph!
Graph this absolute value function: y = – |x + 2| – 3
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?
What is the slope of the right-side ray?

30. Graph this absolute value function: y = – |x + 2| – 3
Let’s graph!
Graph this absolute value function: y = – |x + 2| – 3
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?
What is the slope of the right-side ray?
Use symmetry to draw the left-side ray.

31. Graph this absolute value function: y = – |x + 2| – 3
Let’s graph!
Graph this absolute value function: y = – |x + 2| – 3
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?
What is the slope of the right-side ray?
Use symmetry to draw the left-side ray.
Is this NARROWER, WIDER, or THE SAME as
y = |x| ?

32. Let’s graph!

33. Graph this absolute value function: y = 2/3 |x – 4| + 2
Let’s graph again!
Graph this absolute value function: y = 2/3 |x – 4| + 2

34. Graph this absolute value function: y = 2/3 |x – 4| + 2
Let’s graph again!
Graph this absolute value function: y = 2/3 |x – 4| + 2
Will this graph open UP or DOWN?

35. Graph this absolute value function: y = 2/3 |x – 4| + 2
Let’s graph!
Graph this absolute value function: y = 2/3 |x – 4| + 2
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?

36. Graph this absolute value function: y = 2/3 |x – 4| + 2
Let’s graph again!
Graph this absolute value function: y = 2/3 |x – 4| + 2
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?
What is the slope of the right-side ray?

37. Graph this absolute value function: y = 2/3 |x – 4| + 2
Let’s graph again!
Graph this absolute value function: y = 2/3 |x – 4| + 2
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?
What is the slope of the right-side ray?
Use symmetry to draw the left-side ray.

38. Graph this absolute value function: y = 2/3 |x – 4| + 2
Let’s graph again!
Graph this absolute value function: y = 2/3 |x – 4| + 2
Will this graph open UP or DOWN?
What will be the VERTEX of this graph?
What is the slope of the right-side ray?
Use symmetry to draw the left-side ray.
Is this NARROWER, WIDER, or THE SAME as
y = |x| ?

39. Let’s graph again!

40. Turning graph into a function
We will follow these steps to turn a graph into an absolute value function…
Find the vertex – this gives us h and k.
Find the slope of the right side ray – this gives us a.
Put our h, k, and a into y = a | x – h | + k

41. Turning graph into a function
Let’s turn the following graph into a function!

42. Turning graph into a function
Step 1- Find the vertex (to get h & k)

43. Turning graph into a function
Step 1- Find the vertex (to get h & k) What is the vertex?

44. Turning graph into a function
Step 1- Find the vertex (to get h & k) What is the vertex? (– 4, –1)

45. Turning graph into a function
Step 1- Find the vertex (to get h & k) What is the vertex? (– 4, –1) h = –4 & k = –1

46. Turning graph into a function
Step 2- Find the slope of the right-side ray (to get a)

47. Turning graph into a function
Step 2- Find the slope of the right-side ray (to get a) What is the slope (go from vertex to P1)?

48. Turning graph into a function
Step 2- Find the slope of the right-side ray (to get a) What is the slope (go from vertex to P1)? Slope is –3/2 , so a = –3/2.

49. Turning graph into a function
Step 3- Put our h, k, and a into y = a | x – h | + k

50. Turning graph into a function
Step 3- Put our h, k, and a into y = a | x – h | + k a = –3/2 h = –4 k = –1

51. Turning graph into a function
Step 3- Put our h, k, and a into y = a | x – h | + k a = –3/2 h = –4 k = –1

52. Turning graph into a function
Step 3- y = –3/2 | x – (–4) | + –1 Let’s make this look neater!

53. Turning graph into a function
Step 3- y = –3/2 | x + 4 | – 1 We’re done!