TITLE: What happens to an absolute value graph if $|x|$ has a coefficient
QUESTION [5 upvotes]: I skipped Algebra I in school, and we have Mathematics midterms next week. While going through our review packet, I noticed graphing absolute values, something I had never seen before.
I've figured out the basics: $|x+n|$ translates the graph $n$ units along the x axis, $|x|+d$ translates the graph $d$ units along the y axis, and $-|x|$ flips the graph so it opens downward.
What happens, however, if we have $a|x|$, or $|ax|$? Is there an easy short hand way to draw this, or do I have to make a chart of the points and graph them one by one?

REPLY [3 votes]: I just realized this, moments after posting.
If $a$ exists in the context of the question, it affects the graph as $m$ would in a linear equation.
In $a = 1$ or $a = -1$, the slope on either side is $1$ or $-1$ respectively. The value of $a$ represents the slope on the right hand side of the line of symmetry and the opposite of the slope on the left hand side. (For $y=|ax|$, we can assume $y=|a||x|$, and simply find the abolsute value of $a$, and get $y=a|x|$)
For example, if $a=1$:

But, if $a=2$,