​$$y=\vert f(x)\vert$$​ and $$y=f(\vert x \vert)$$​​

​​In a nutshell

There is a difference between $$y=\vert f(x)\vert$$ and $$y=f(\vert x \vert)$$. The first modulus function simply gives the output as a positive number, whereas the second function forces the value of $$x$$​ to be positive but not the output.

​$$y=\vert f(x)\vert$$​​

This modulus function creates a reflection of all points under the $$x$$​-axis because all $$y$$ values are positive. To sketch it you can simply follow these steps:

Procedure

Example 1

On the same set of axes, sketch $$f(x)=x+2$$ and $$f(x)=\vert x+2\vert$$.

You have to sketch your normal $$f(x)$$​ but when the values of $$y$$ are negative, reflect them over the positive axis:

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$$y=f(\vert x \vert)$$​​

This function stays the same when $$x\ge0$$​ and creates a reflection in the $$y$$-axis. To sketch it you can simply follow these steps:

procedure

Example 2

On the same set of axes, sketch $$f(x)=2x-2$$ and $$f(x)=2\vert x\vert-2$$.

You have to sketch your normal $$f(x)=2x-2$$​ but mirror it on the negative side of the $$x$$​-axis:

Example 3

Sketch the following functions on the same set of axes:

Your graphs should look like this:

As you can see, $$f(\vert x\vert)$$​ mirrors the $$y$$​-axis and $$\vert f(x)\vert$$​ reflects off the $$x$$-axis.

Example 4

Sketch the following functions on the same set of axes:

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Your three functions should look like this: