Special Graphs

Graphing 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.