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Absolute Value Functions

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Presentation on theme: "Absolute Value Functions"— Presentation transcript:

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!

54


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