# Describing the Graph of a Function

Sometimes you need to describe the graph of a function in a non-symbolic way. For example, you may be asked

- whether a function is increasing or decreasing;
- whether it has one minimum value or maximum value, or several such values
- whether it is linear or not
- whether the rate of change is constant, increasing, or decreasing
- whether it has an upper or lower bound.

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Example 1:
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Describe the two functions $f\left(x\right)$ and $g\left(x\right)$ , using the terms increasing, decreasing, maxima and minima.

The graph of $f\left(x\right)$ is periodic. It decreases for $-3<x<-1$ , then increases for $-1<x<1$ , then decreases again for $1<x<3$ , etc.

It has a maximum value of $1$ and a minimum value of $-1$ , and it attains these maxima and minima many times. The upper bound of the function is $1$ and the lower bound is $-1$ .

The graph of $g\left(x\right)$ is increasing for $-\infty <x<-1$ and decreasing for $-1<x<\infty $ . The graph takes a maximum value of $3$ at $x=-1$ . It has no minimum.

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Example 2:
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Which graph has a greater rate of change?

Both graphs start together at $\left(0,0\right)$ . At first, the linear function, $g\left(x\right)$ , has the faster rate of change.

But $f\left(x\right)$ soon catches up, and surpasses $g\left(x\right)$ at $\left(8,16\right)$ , and continues increasing at a faster rate.