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Graphing Absolute Value and Cubic Functions

Graphing Absolute Value and Cubic Functions

Graphing Absolute Value and Cubic Functions

Graphing Absolute Value and Cubic Functions

Graphing Absolute Value and Cubic Functions

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| :

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 :

f (x) = 2| x - 1| - 4
Here is the graph of f (x) = - | x + 2| + 3 :
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 :

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 :
y = (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.