Graph the function. Label all intercepts, critical points, and inflection points.

Answer

• Find dots.

The first thing to do is factor the numerator of *f*:

Now we can see that *f* is undefined with a vertical asymptote at *x* = -2, and has roots at *x* = 0 and *x* = 1.

The derivative of *f* is

This is zero when the numerator is zero, which occurs at (using the quadratic formula)

We have critical points at

The second derivative of *f* is

This is never 0, and is undefined at the same place *f* is undefined, so the function *f* has no inflection points.

Here are the dots:

• Find shapes.

Here's the numberline:

First, find the sign of the function .

When *x* < -2 all quantities *x*, ( *x* - 1 ), and (x + 2) are negative, so *f* is negative.

When -2 < *x* < 0 the quantities in the numerator are both negative and the denominator is positive, so *f* is positive.

When 0 < *x* < 1 the quantity *x* - 1 is negative, while *x* and *x* + 2 are positive, so *f* is negative.

When 1 < *x* all quantities *x*, ( *x* - 1 ), and (x + 2) are positive, so *f* is positive.

Next, the sign of .

The denominator is always positive, so we don't need to worry about that. Since f'(0) is negative, the derivative *f*' is negative in between the two critical points. Plugging in some other numbers, we can see what the derivative is doing outside the critical points.

and

The derivative *f*' is positive outside the critical points.

Finally, the sign of . This one is negative when *x* < -2 and positive when *x* > -2.

Using this, we find the shapes of *f*:

• Play connect-the-dots.

Here's the final picture.