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Course: Algebra (all content) > Unit 8

Lesson 1: Graphs of absolute value functions
Math
Algebra (all content)
Absolute value equations, functions, & inequalities
Graphs of absolute value functions
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Absolute value graphs review

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The general form of an absolute value function is f(x)=a|x-h|+k. From this form, we can draw graphs. This article reviews how to draw the graphs of absolute value functions.
General form of an absolute value equation:
f(x)=a|x−h|+k
The variable a tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables h and k tell us how far the graph shifts horizontally and vertically.
Some examples:
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals the absolute value of x. The vertex is at the point zero, zero. The points negative one, one and one, one can be found on the graph.
Graph of y=|x|
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals three times the absolute value of x. The vertex is at the point zero, zero. The points negative one, three and one, three can be found on the graph.
Graph of y=3|x|
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals negative one times the absolute value of x. The vertex is at the point zero, zero. The points negative one, negative one and one, negative one can be found on the graph.
Graph of y=-|x|
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals the absolute value of the sum of x plus three minus two. The vertex is at the point negative three, negative two. The points negative two, negative one and negative four, negative one can be found on the graph.
Graph of y=|x+3|-2

Example problem 1

We're asked to graph:
f(x)=|x−1|+5
First, let's compare with the general form:
f(x)=a|x−h|+k
The value of a is 1, so the graph opens upwards with a slope of 1 (to the right of the vertex).
The value of h is 1 and the value of k is 5, so the vertex of the graph is shifted 1 to the right and 5 up from the origin.
Finally here's the graph of y=f(x):
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals the absolute value of the difference of x minus one plus five. The vertex is at the point one, five. The points zero, six and two, six can be found on the graph.

Example problem 2

We're asked to graph:
f(x)=−2|x|+4
First, let's compare with the general form:
f(x)=a|x−h|+k
The value of a is −2, so the graph opens downwards with a slope of −2 (to the right of the vertex).
The value of h is 0 and the value of k is 4, so the vertex of the graph is shifted 4 up from the origin.
Finally here's the graph of y=f(x):
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals negative two times the absolute value of x plus four. The vertex is at the point zero, four. The points negative one, two and one, two can be found on the graph.
Want to learn more about absolute value graphs? Check out this video.
Want more practice? Check out this exercise.

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