Graphing Absolute-Value Functions

• Graph y = | x + 2 |

One of the other students does what is commonly done: he picks only positive x-values for his T-chart:

Then he plots his points:

These points are fine, as far as they go, but they aren't enough; they don't give an accurate idea of what the graph should look like. In particular, they don't include any "minus" inputs, so it's easy to forget that those absolute-value bars mean something. As a result, the student forgets to take account of those bars, and draws an erroneous graph:

Aaaaaand... he just flunked the quiz.

But you're more careful. You remember that absolute-value graphs involve absolute values, and that absolute values affect "minus" inputs. So you pick x-values that put a "minus" inside the absolute value, and you choose quite a few more points. Your T-chart looks more like this:

Then you plot your points:

...and finally you connect your dots:

You have the correct graph:

Aaaaand... you just aced the quiz. Good work!

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While absolute-value graphs tend to look like the one above, with an "elbow" in the middle, this is not always the case. However, if you see a graph with an elbow like this, you should expect that the graph's equation probably involves an absolute value. In all cases, you should take care that you pick a good range of x-values, because three x-values right next to each other will almost certainly not give you anywhere near enough information to draw a valid picture.

Note: The absolute-value bars make the entered values evaluate to being always non-negative (that is, positive or zero). As a result, the "V" in the above graph occurred where the sign on the inside was zero. When x was less than −2, the expression x + 2 was less than zero, and the absolute-value bars flipped those "minus" values from below the x-axis to above it. When x equalled −2, then the argument (that is, the expression inside the bars) equalled zero. For all x-values to the right of −2, the argument was positive, so the absolute-value bars didn't change anything.

In other words, graphically, the absolute-value bars took this graph:

...and flipped the "minus" part (in green on the graph) from below the x-axis to above it. Noticing where the argument of the absolute-value bars will be zero can be helpful in ensuring that you're doing the graph correctly.

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• Graph y = | x | + 2

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Because absolute-value bars always spit out non-negative values, it can be tempting to assume that absolute-value graphs can not go below the x-axis. But they can:

• Graph y = −| x + 2 |

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Also, don't assume that any absolute-value graph will be only ever on one side of the x-axis. The graphs can cross:

• Graph y = −| x | + 2