Absolute Value Part 1
y = a|x - h| + k

Enter a = 1, h = 0, k = 0
gives y = 1| x - 0|+ 0  or y = | x |

Click Graph, then Click Store.
Use this graph to compare

1 . Enter a = 1, h = 0, k = -2

Predict which way the graph will SHIFT?
 Up Left None Down Right
Now click Graph.

2. Enter a = 1, h = 0, k = 1

Predict which way the graph will SHIFT?
 Up Left None Down Right
Now click Graph.

3. Enter a = 1, h = 0, k = -5/2

Predict which way the graph will SHIFT?
 Up Left None Down Right
Now click Graph.
Absolute Value Part 2
y = a|x - h| + k

Enter a = 1, h = 0, k = 0
gives y = 1| x - 0| + 0  or y = | x |

Click Graph, then Click Store.
Use this graph to compare

1 . Enter a = 1, h = 3, k = 0

Predict which way the graph will SHIFT?
 Up Left None Down Right
Now click Graph.

2. Enter a = 1, h = -1, k = 0

Predict which way the graph will SHIFT?
 Up Left None Down Right
Now click Graph.

3. Enter a = 1, h = 5/3, k = 0

Predict which way the graph will SHIFT?
 Up Left None Down Right
Now click Graph.
Absolute Value Quick Quiz
Without graphing predict which way the graph will shift from  y = | x|

1.  y = |x + 2|

Predict which way the graph will SHIFT?
 Up 2 Left 2 None Down 2 Right 2
Verify with grapher on the left.

2.  y = | x| + 3

Predict which way the graph will SHIFT?
 Up 3 Left 3 None Down 3 Right 3
Verify with grapher on the left.

3.  y = |x - 3|

Predict which way the graph will SHIFT?
 Up 3 Left 3 None Down 3 Right 3
Verify with grapher on the left.

4.  y = | x| - 1

Predict which way the graph will SHIFT?
 Up 1 Left 1 None Down 1 Right 1
Verify with grapher on the left.
Absolute Value Part 3
y = a|x - h| + k

Enter a = 1, h = 0, k = 0
gives y = 1| x - 0| + 0  or y = | x |

Click Graph, then Click Store.
Use this graph to compare

1 . Enter a = 1/3, h = 0, k = 0

Predict which way the graph will CHANGE?
 Vertical Shrink - widen None Vertical Stretch - narrow
Now click Graph.

2. Enter a = 3, h = 0, k = 0

Predict which way the graph will CHANGE?
 Vertical Shrink - widen None Vertical Stretch - narrow
Now click Graph.

3. Enter a = -2, h = 0, k = 0

Predict which way the graph will CHANGE?
 Reflect across  x -axis None Reflect across  y -axis
Now click Graph.

4. Enter a = 1/10, h = 0, k = 0

Predict which way the graph will CHANGE?
 Vertical Shrink - widen None Vertical Stretch - narrow
Now click Graph.

5. Enter a = -1, h = 0, k = 0

Predict which way the graph will CHANGE?
 Reflect across  x -axis None Reflect across  y -axis
Now click Graph.
Absolute Value Altogether
Predict the following shifting, then verify using grapher.
y = 2| x - 3 |  + 4
y = -2| x + 3 |  - 4
y = 1/4| x + 2 |  + 1
y = -4| x - 1 |  -  2
y = | x - 2/5 | - 1/5

Absolute Value Part 4
y = a|x - h| + k
Although most books won't call the point (h,k) the vertex (as in quadratics) but it will either be a maximum or minimum value for the absolute value function.
If a > 0 , then (h,k) is the lowest point of the graph.  (opens upward)
If a < 0 , then (h,k) is the highest point of the graph.  (opens downward)

Examples:

Absolute Value Min or Max
y = |x - 1/2 | + 3/5 (1/2, 3/5) Min
y = -2 |x + 3 | - 4 (-3,-4) Max
y = 3|x + 1|   (-1,0) Min
y = -2|x|  - 4 (0,-4) Max
y = 2 |x - 3|  + 4 (3,4) Min