Taking the absolute value of a negative number makes it positive. For this reason, graphs of absolute value functions tend not to look quite like the graphs of linear functions that you've already studied. However, because of how absolute values behave, it is important to include negative inputs in your T-chart when graphing absolute-value functions. If you do not pick x-values that will put negatives inside the absolute value, you will usually mislead yourself as to what the graph looks like.
For instance, suppose your class is taking the following quiz:
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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:
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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:
WRONG ANSWER!
Aaaaaand... he just flunked the quiz.
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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:
Right answer!
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.
This function is almost the same as the previous one.
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However, the argument of the previous absolute-value expression was x + 2. In this case, only the x is inside the absolute-value bars. This argument will be zero when x = 0, so I should expect to see an elbow in that area. Also, since the "plus two" is outside of the absolute-value bars, I expect my graph to look like the regular absolute-value graph (being a "V" with the elbow at the origin), but moved upward by two units.
First, I'll fill in my T-chart, making sure to pick some negative x-values as I go:
Then I'll draw my dots and fill in the graph:
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:
This function is kind of the opposite of the first function (above), because there is a "minus" on the absolute-value expression on the right-hand side of the equation. Because of this "minus", the positive values provided by the absolute-value bars will all be switched to negative values. In other words, I should expect this graph to have its elbow at (−2, 0) like the first graph above, but the rest of the graph will be flipped upside down to be below the x-axis.
First, I'll fill in my T-chart:
Then I do my graph:
Also, don't assume that any absolute-value graph will be only ever on one side of the x-axis. The graphs can cross:
My T-chart:
...and my graph:
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