Answer
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_33.png)
• Find dots.
The first thing to do is factor the numerator of f:
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_34.png)
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
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_35.png)
This is zero when the numerator is zero, which occurs at (using the quadratic formula)
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_36.png)
We have critical points at
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_37.png)
The second derivative of f is
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_38.png)
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:
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_graphik_43.png)
• Find shapes.
Here's the numberline:
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_graphik_44.png)
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.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_graphik_45.png)
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.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_41.png)
and
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_latek_42.png)
The derivative f' is positive outside the critical points.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_graphik_46.png)
Finally, the sign of
. This one is negative when x < -2 and positive when x > -2.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_graphik_47.png)
Using this, we find the shapes of f:
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_graphik_48.png)
• Play connect-the-dots.
Here's the final picture.
![](https://media1.shmoop.com/images/calculus/calc_hghderiv_graphs_narr_graphik_49.png)