1. Absolute Value Functions

2. The Absolute Value Function
The main objective of this lesson is to learn about the absolute value function
*The general form or pattern for an absolute value function is 𝑓 𝑥 = 𝑎 𝑥−ℎ +𝑘, where 𝑎, ℎ, 𝑘 are constants and 𝑎≠0
*The function obtained when 𝑎=1 and both ℎ and 𝑘 are zero is sometimes called the parent function
The parent function is 𝑓 𝑥 =1⋅ 𝑥−0 +0
*Because of the Identity Properties of Addition and Multiplication, we can write the parent function as 𝑓 𝑥 = 𝑥

3. Domain of the Absolute Value Parent Function
What is the domain of the absolute value parent function?
We can determine this by looking at the piecewise definition of absolute value
𝑓 𝑥 = 𝑥, if 𝑥≥0 −𝑥, if 𝑥<0
Note that the inequalities, together, are the same as −∞,0 ∪[0,∞)
*What is the graph of this compound inequality?
*What is the domain of the absolute value parent function?

4. The Range of the Absolute Value Parent Function
Recall that the absolute value was defined so that it always returned a positive value (or zero)
In other words, 𝑥 ≥0 for any value of 𝑥 from the domain
But since 𝑦=|𝑥|, then 𝑦≥0; this is the range of the parent function
*In interval notation this is: _______________________
*In set-builder notation this is: ___________________

5. The Graph of the Absolute Value Function

6. The Graph of the Absolute Value Function
Note that the graph of the absolute value parent function is v-shaped; this will be the case for all absolute value functions
*The tip of the v shape is called the vertex of the function
*The goal here is for you to be able to graph any absolute value function by varying the values of 𝑎, ℎ, 𝑘 in 𝑓 𝑥 =____________, or to predict what the graph will look like
We can also use the graph of the function (of any function, in fact) to understand how the solutions of equations relate to the function itself

7. The Graph of the Absolute Value Function
You will next see an animation that will allow you to see how the graph of the absolute value function changes if we change the values of 𝑎,ℎ,𝑘 in 𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘. Answer the following questions as you view the animation
*How is the graph of 𝑓 𝑥 =|𝑥−ℎ| transformed if:
ℎ is positive:____________________________________________________
ℎ is negative:____________________________________________________

8. The Graph of the Absolute Value Function
*How is the graph of 𝑓 𝑥 = 𝑥 +𝑘 transformed if:
𝑘 is positive:____________________________________________________
𝑘 is negative:____________________________________________________

9. The Graph of the Absolute Value Function
*How is the graph of 𝑓 𝑥 =𝑎 𝑥 transformed if:
𝑎 >1:____________________________________________________
𝑎 <1:____________________________________________________
𝑎>0:_____________________________________________________
𝑎<0:_____________________________________________________

10. How to Graph the Absolute Value Function
We can always graph any function by finding pairs 𝑥,𝑦 and plotting them in the coordinate plane
*But note that the absolute value function is like parts of two lines, one of which has a positive slope, the other of which has a negative slope
*Each of these lines has a “slope” value; but rather than the 𝑦-intercept as a start value, we will use the vertex
So, our first task is to determine how to use the absolute value function to find the vertex
To do this, we will again turn to the definition of the absolute value

11. How to Graph the Absolute Value Function
Recall the definition (I’ve changed the variable to avoid confusion):
𝑟 = 𝑟, if 𝑟≥0 −𝑟, if 𝑟<0
The general pattern for our function is 𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘; apply the definition to this:
𝑓 𝑥 = 𝑎 𝑥−ℎ +𝑘, if 𝑥−ℎ≥0 −𝑎 𝑥−ℎ +𝑘, if 𝑥−ℎ<0
We can think of the far right inequalities as the domain for each part of the function; we now solve these for 𝑥

12. How to Graph the Absolute Value Function
𝑓 𝑥 = 𝑎 𝑥−ℎ +𝑘, if 𝑥≥ℎ −𝑎 𝑥−ℎ +𝑘, if 𝑥<ℎ
This shows us that the function “divides” at the value of ℎ on the 𝑥- axis
This means that our vertex point is (ℎ,?); to find the 𝑦-coordinate, substitute ℎ for 𝑥:
𝑓 ℎ =𝑎 ℎ−ℎ +𝑘=𝑎⋅0+𝑘=𝑘
*Hence, in the absolute value function 𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘, the vertex is at the point__________, which we can read from the function itself

13. How to Graph the Absolute Value Function
*Example: What are the coordinates of the vertex point for 𝑓 𝑥 = 2 𝑥−1 +2?__________
Graph the function on your calculator to verify your answer
*Example: What are the coordinates of the vertex point for 𝑔 𝑥 = −3 𝑥+2 −1? __________

14. How to Graph the Absolute Value Function
*Example: What are the coordinates of the vertex point for ℎ 𝑥 = 1 3 𝑥 +1? __________
Graph the function with your calculator to verify your answer.
*Example: What are the coordinates of the vertex point for 𝑘 𝑥 =− 1 2 |𝑥−2|?__________

15. How to Graph the Absolute Value Function
From our previous work, we see that the graph of an absolute value function is part of a linear function to the right of the vertex, and part of another linear function to the left of the vertex
These are:
𝑎 𝑥−ℎ +𝑘, if 𝑥≥ℎ
−𝑎 𝑥−ℎ +𝑘, if 𝑥<ℎ
In the first case, 𝑎 is the slope of the partial line (a ray, in fact), and −𝑎 is the slope of the other partial line (also a ray)
Once you have located the vertex point, find a few other points to right and left of the vertex by using each “slope”

16. How to Graph the Absolute Value Function
Here is an example: graph the absolute value function 𝑓 𝑥 = 𝑥−2 +1
What is the vertex point? __________
What is the right-slope? __________
What is the left-slope? __________
Now plot the vertex point and graph the function

17. How to Graph the Absolute Value Function

18. How to Graph the Absolute Value Function
Example: graph the absolute value function 𝑓 𝑥 =2 𝑥 −1
What is the vertex point? __________
What is the right-slope? __________
What is the left-slope? __________
Now plot the vertex point and graph the function

19. How to Graph the Absolute Value Function

20. How to Graph the Absolute Value Function
Example: graph the absolute value function 𝑓 𝑥 = 1 3 𝑥+1
What is the vertex point? __________
What is the right-slope? __________
What is the left-slope? __________
Now plot the vertex point and graph the function

21. How to Graph the Absolute Value Function

22. Guided Practice Graph the following absolute value functions. 𝑦= 𝑥−1
𝑓 𝑥 =2 𝑥
𝑔 𝑥 =−2 𝑥+1 −2
ℎ 𝑥 =3 𝑥−2 +1

23. Concentrate!