Graphs of f ( x ) and f ' ( x ) Examples

From the graph of f(x), draw a graph of its derivative f ' (x).

Since f is a line, its slope is constant. Moreover Since f is sloping upward, its slope is a positive constant. This means f ' (x) is a positive constant function:

From the graph of f(x), draw a graph of f ' (x).

The slope of f starts out negative, and gets closer to zero as we move to the right, and then settles at zero:

This means f ' ( x) will start out negative, approach 0, and then remain at 0 from some point onwards:

From the graph of f(x), draw a graph of f ' (x).

We can see that f starts out with a positive slope (derivative), then has a slope (derivative) of zero, then has a negative slope (derivative):

This means the derivative will start out positive, approach 0, and then become negative:

Be Careful: Label your graphs f or f ' appropriately. When we're graphing both functions and their derivatives, it can be confusing to remember which graph we're looking at.

From the graph of f ' (x), draw a graph of f(x).

Since f ' is zero everywhere, the slope of f is zero everywhere, so f must be constant. f could be a positive constant:

or zero:

or a negative constant:

As we can see from the previous example, we can't tell, given a graph of f', exactly what f will look like. We can only tell the general "shape" of f: that is, where f increases and where it decreases.

From the graph of f ' (x), draw a graph of f(x).

f ' is negative, then zero, then positive:

This means f will be decreasing for a bit, and will then turn around and increase:

We just don't know exactly where the graph of f(x) will be in relation to the y-axis. We can figure out the general shape of f, but f  we could take the graph of f that we just made and shift it up or down along the y-axis to anywhere we want. Our answer would still be correct.

From the graph of f ' (x), draw a graph of f(x).

First f ' is negative, then positive, then zero:

This means f will be decreasing, then flatten out, and then start increasing.