SummaryGraphing Absolute Value and Cubic Functions

Graphing the Absolute Value Function

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|:

The graph looks like a "V", with its vertex at
(0, 0). Its slope is m = 1 on the
right side of the vertex, and m = - 1 on the left side of the vertex.

We can translate, stretch, shrink, and reflect the graph.
Here is the graph of f (x) = 2| x - 1| - 4:

Here is the graph of f (x) = - | x + 2| + 3:
In general, the graph of the absolute value function f (x) = a| x - h| + k is a
"V" with vertex (h, k), slope m = a on the right side of the vertex (x > h) and slope m = - a on the left side of the vertex (x < h). The graph of
f (x) = - a| x - h| + k is an upside-down "V" with vertex (h, k), slope m = - a for x > h and slope m = a for x < h.

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.

Graphing the Cubic Function

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

We can translate, stretch, shrink, and reflect the graph of f (x) = x^{3}. Here is the
graph of f (x) = (x - 2)^{3} + 1:

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.