Again, there’s no x-value where this is zero. In fact, the second derivative is positive at all values of x
except 0. Therefore, the graph is concave up everywhere.
Now it’s time to graph this.
There’s one other type of graph you should know about: a rational function. In order to graph a rational
function, you need to know how to find that function’s asymptotes.
How to Find Asymptotes
A line y = c is a horizontal asymptote of the graph of y = f(x) if
f(x) = c or if f(x) = cA line x = k is a vertical asymptote of the graph of y = f(x) if
f(x) = ±∞ or if f(x) = ±∞Example 4: Sketch the graph of y = .
Step 1: Find the x-intercepts. A fraction can be equal to zero only when its numerator is equal to zero
(provided that the denominator is not also zero there). All we have to do is set 3x = 0, and you get x = 0.
Thus, the graph has an x-intercept at (0, 0). Note: This is also the y-intercept.
Next, look for asymptotes. The denominator is undefined at x = −2, and if we take the left- and right-hand
limits of the function, we see the following: