1. Use Absolute Value Functions & Transformations Part II
Chapter 2.7

2. Changing Parameter a
The general form of an absolute value function is 𝑦=𝑎 𝑥−ℎ +𝑘
As we have previously seen, changing parameter h causes the graph of the parent function to shift either right or left
Changing parameter k causes the graph of the parent function to shift either up or down
What effect does changing a have?
Before we investigate graphically, let’s try to make some sense out of what we already know

3. Changing Parameter a
Recall the definition of absolute value 𝑥 = 𝑥, if 𝑥>0 0, if 𝑥=0 −𝑥, if 𝑥<0
Applying the definition to the general form gives us the following

4. Changing Parameter a
𝑎 𝑥−ℎ +𝑘= 𝑎 𝑥−ℎ +𝑘, if 𝑥>ℎ 𝑘, if 𝑥=ℎ 𝑎 −𝑥+ℎ +𝑘, if 𝑥<ℎ These are parts of two lines, one with positive slope and the other with negative slope: 𝑦=𝑎𝑥−𝑎ℎ+𝑘 and 𝑦=−𝑎𝑥+𝑎ℎ+𝑘 The slope of the first line is the same as the value of a, and the slope of the second line is the value of −𝑎. We can use these to graph the absolute value function after plotting the vertex point.

5. Changing Parameter a
For example, suppose we want to graph the absolute value function 𝑓 𝑥 =2 𝑥−1 −3
As before, the vertex point is at (1,−3), and this divides the graph into two parts, each part of a line
The two lines are 𝑦=2 𝑥−1 −3=2𝑥−5, if 𝑥>1 and 𝑦=2(−𝑥+1)−3=−2𝑥−1, for 𝑥<1
The graphs of each line are shown on the next slide

6. Changing Parameter a

7. Changing Parameter a
The graph shows us that we can graph absolute value functions by using the parameter a as a slope
How does changing parameter a affect the shape of the graph of the parent function?
Go to the applet at and follow the directions
Be sure to complete the statements at the bottom, copied into your notes

8. Graphing an Absolute Value Function
Graph the absolute value function 𝑦=3 𝑥−2 −4 and compare to the parent function, 𝑦=|𝑥|.

9. Graphing an Absolute Value Function
Graph the absolute value function 𝑦=3 𝑥−2 −4 and compare to the parent function, 𝑦=|𝑥|.
Note that the vertex point is at 2,−4 and we use slope to find a second point
The line of symmetry is the vertical line 𝑥=2

10. Graphing an Absolute Value Function

11. Guided Practice
Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|.
𝑦=2 𝑥−1 −3
𝑦=− 𝑥+2
𝑦= 1 2 𝑥 +1
𝑦= −2 3 𝑥+1 +2
𝑓 𝑥 =−2 𝑥−3 +1

12. Guided Practice
Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|.
𝑦=2 𝑥−1 −3

13. Guided Practice
Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|.
𝑦=− 𝑥+2

14. Guided Practice
Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|.
𝑦= 1 2 𝑥 +1

15. Guided Practice
Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|.
𝑦= −2 3 𝑥+1 +2

16. Guided Practice
Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|.
𝑦=−2 𝑥−3 +1

17. Exercise 2.7b
Page 127, #3-20