The graph of the absolute value function
*f* (*x*) = | *x*|
is similar to the graph of
*f* (*x*) = *x*
except that the "negative" half of
the graph is reflected over the
*x*
-axis. Here
is the graph of
*f* (*x*) = | *x*|
:

We can translate, stretch, shrink, and reflect the graph.

Here is the graph of
*f* (*x*) = 2| *x* - 1| - 4
:

If
*a* > 0
, then the lowest
*y*
-value for
*y* = *a*| *x* - *h*| + *k*
is
*y* = *k*
. If
*a* < 0
, then the greatest
*y*
-value for
*y* = *a*| *x* - *h*| + *k*
is
*y* = *k*
.

Here is the graph of
*f* (*x*) = *x*
^{3}
:

In general, the graph of
*f* (*x*) = *a*(*x* - *h*)^{3} + *k*
has vertex
(*h*, *k*)
and is
stretched by a factor of
*a*
. If
*a* < 0
, the graph is
reflected over the
*x*
-axis.

Take a Study Break!