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)=axh+kf(x)=\goldD{a}|x-\blueD{h}|+\blueD{k}
The variable a\goldD{a} tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables h\blueD h and k\blueD k tell us how far the graph shifts horizontally and vertically.
Some examples:
Graph of y=|x|
Graph of y=3|x|
Graph of y=-|x|
Graph of y=|x+3|-2

Example problem 1

We're asked to graph:
f(x)=x1+5f(x)=|x-1|+5
First, let's compare with the general form:
f(x)=axh+kf(x)=\goldD{a}|x-\blueD{h}|+\blueD{k}
The value of a\goldD a is 11, so the graph opens upwards with a slope of 11 (to the right of the vertex).
The value of h\blueD h is 11 and the value of k\blueD k is 55, so the vertex of the graph is shifted 11 to the right and 55 up from the origin.
Finally here's the graph of y=f(x)y=f(x):

Example problem 2

We're asked to graph:
f(x)=2x+4f(x)=-2|x|+4
First, let's compare with the general form:
f(x)=axh+kf(x)=\goldD{a}|x-\blueD{h}|+\blueD{k}
The value of a\goldD a is 2-2, so the graph opens downwards with a slope of 2-2 (to the right of the vertex).
The value of h\blueD h is 00 and the value of k\blueD k is 44, so the vertex of the graph is shifted 44 up from the origin.
Finally here's the graph of y=f(x)y=f(x):
Want to learn more about absolute value graphs? Check out this video.
Want more practice? Check out this exercise.
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