Example 2: Sketch the graph of y = x^4 + 2x^3 − 2x^2 + 1.
Step 1: First, let’s find the x-intercepts.
x^4 + 2x^3 − 2x^2 + 1 = 0If the equation doesn’t factor easily, it’s best not to bother to find the function’s roots. Convenient, huh?
The good news is that if the roots aren’t easy to find, ETS won’t
ask you to find them, or you can find them with your calculator.Next, let’s find the y-intercepts.
y = (0)^4 + 2(0)^3 − 2(0)^2 + 1The curve has a y-intercept at (0, 1).
There are no vertical asymptotes because there is no place where the curve is undefined.
Step 2: Now we take the derivative to find the critical points.
= 4x^3 + 6x^2 − 4x