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But what about the shape of the graph? The graph shows a gradual incline up to 18", and then a precipitous drop back down to 12"; and this pattern repeats throughout the shown time. The most likely explanation is that Alice’s hair grows slowly until it reaches 18", at which point she goes to the hair stylist and has it cut down, within a very short time (an hour or so), to 12". Then the gradual growth begins again.
Consider the following graph.
This is our earlier “U” shaped graph ( $y={x}^{2}$ ) turned on its side. This might seem like a small change. But ask this question: what is $y$ when $x=3$ ? This question has two answers. This graph contains the points $(\mathrm{3,}-9)$ and $(\mathrm{3,9})$ . So when $x=3$ , $y$ is both 9 and –9 on this graph.
This violates the only restriction on functions—the rule of consistency . Remember that the $x$ -axis is the independent variable, the $y$ -axis the dependent. In this case, one “input” value $(3)$ is leading to two different “output” values $(-\mathrm{9,9})$ We can therefore conclude that this graph does not represent a function at all. No function, no matter how simple or complicated, could produce this graph.
This idea leads us to the “vertical line test,” the graphical analog of the rule of consistency.
It is important to understand that the vertical line test is not a new rule! It is the graphical version of the rule of consistency. If any vertical line touches a graph in two places, then the graph has two different $y$ -values for the same $x$ -value, and this is the only thing that functions are not allowed to do.
Suppose the following is the graph of the function $y=f(x)$ .
We can see from the graph that the domain of the graph is $-3\le x\le 6$ and the range is $-3\le y\le 2$ .
Question: What does the graph of $y=f(x)+2$ look like ?
This might seem an impossible question, since we do not even know what the function $f(x)$ is. But we don’t need to know that in order to plot a few points.
$x$ | $f(x)$ | $f(x+2)$ | so $y=f(x)$ contains this point | and $y=f(x)+2$ contains this point |
---|---|---|---|---|
–3 | 2 | 4 | $(-\mathrm{3,2})$ | $(-\mathrm{3,4})$ |
–1 | –3 | –1 | $(-\mathrm{1,}-3)$ | $(-\mathrm{1,}-1)$ |
1 | 2 | 4 | $(\mathrm{1,2})$ | $(\mathrm{1,4})$ |
6 | 0 | 2 | $(\mathrm{6,0})$ | $(\mathrm{6,2})$ |
If you plot these points on a graph, the pattern should become clear. Each point on the graph is moving up by two . This comes as no surprise: since you added 2 to each y-value, and adding 2 to a y-value moves any point up by 2. So the new graph will look identical to the old, only moved up by 2.
In a similar way, it should be obvious that if you subtract 10 from a function, the graph moves down by 10. Note that, in either case, the domain of the function is the same, but the range has changed.
These permutations work for any function . Hence, given the graph of the function $y=\sqrt{x}$ below (which you could generate by plotting points), you can produce the other two graphs without plotting points, simply by moving the first graph up and down.
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