MM - Absolute Value Functions Lesson

Absolute Value Functions

The following step-function can be written as a piece-wise function. 

$f\left(x\right)= \begin{cases} -3, & x<-2\\ 0, & -2\le x\le 1\\ 3, & x> 1 \end{cases}$

The graph below shows the piecewise function.

$f\left(x\right)= \begin{cases} x^2, & \text{if }x<2\\ 6, & \text{if }x=2\\ -x+10, & \text{if }x> 2 \end{cases}$

The graph below shows an absolute value function and the equation is written as a piecewise function.

$f\left(x\right)= \begin{cases} x, & \text{if }x>0\\ -x, & \text{if }x<0\\ \end{cases}$

The third new type of function (which could also be written as a piecewise function) is an absolute value function. Recall the absolute value of a number is the distance the number is from zero on the number line, which is always positive. Therefore, the parent absolute value function will have a range consisting of zero and positive numbers.

Here is an example:  $f\left(x\right)=\left|x+2\right|$ .  

To evaluate  $f\left(-5\right)$ , put -5 in for x.

|-5+2| = |-3| = 3 so f(-5) = 3

The graph of an absolute value function has a V-shape. The standard form for an absolute value function is  $y=a\left|x-h\right|+k$  , where (h, k) is the vertex.  To graph an absolute value function, find the vertex and then pick x values to the left and right of the vertex and substitute those values in for x to get a corresponding y-value. Then plot the points and draw the graph.

 

Watch the following videos that will explore absolute value functions.

IMAGES CREATED BY GAVS