Graphing Absolute Value Functions

Let's draw the graph of $y=|x|,$ and divide it into two parts. One part for positive $x$-values and one for negative.

Positive $x$-values and $x=0$

When taking the absolute value of a positive number, the absolute value will not affect anything. This means that when $x$ is greater than $0,$ the output of the function will be the same as the input value. Therefore, the graph of $y=|x|$ is identical to the one for $$y=x\text{when }x\ge 0.$$ For $x=0,$ the output of the function is $0.$ This means that the origin is a point on the graph.

The negative $x$-values

When we have negative input values, the absolute value makes the value change its sign and the output becomes positive. This sign change can be represented by a minus sign in front of the function we drew for positive $x$-values, which means $$y=\text{-}x\text{when }x\le 0.$$ For negative $x$-values, we mirror the graph of $y=x$ to get $y=|x|.$ The graph will then be placed above the $x$-axis at the corresponding positive numbers.

The output of the expression $|x|$ can only be positive numbers and $0.$ But the minus sign in front of $|x|$ will make the positive values negative. Let's study this by looking at a few examples.

For the function $y=\text{-}|x|$, this means that all values of the function change sign and end up under the $x$-axis at the corresponding negative value.

Note that this function is not given in the form $y=|f(x)|$ but on the form $y=\text{-}|f(x)|,$ which means that the graph in this case ends up below the $x$-axis.