= −∞ andThe curve has a vertical asymptote at x = −2.
If we take = 3 and = 3, the curve has a horizontal asymptote at y = 3.
Step 2: Now, take the derivative to figure out the critical points.
=
There are no values of x that make the derivative equal to zero. Because the numerator is 6 and the
denominator is squared, the derivative will always be positive (the curve is always rising). You should
note that the derivative is undefined at x = −2, but you already know that there’s an asymptote at x = −2, so
you don’t need to examine this point further.
Step 3: Now, it’s time for the second derivative.
This is never equal to zero. The expression is positive when x < −2, so the graph is concave up when x <
−2. The second derivative is negative when x > −2, so it’s concave down when x > −2.
Now plot the graph.