Page 46 - Calc for the Clueless

The intercept is (0,8). Vertical asymptotes: bottom of fraction = 0. x =-2 and x = 1. Horizontal asymptote: if we
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multiplied out the top (something you would never do), the highest power of x on top is X . On the bottom is
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X . The degree of the top is less than the bottom.
The horizontal asymptote is y = 0. Oblique asymptote: none, since there is a horizontal one.

We are ready to start the sketch. It is advisable to use three colors, one for the axes, one for the asymptotes, and
one for the sketch.

We need to substitute only one number!!! That number is to the right of the rightmost vertical asymptote or x
+
intercept. f(2 ) is positive. Since the power of (x - 2) is even, namely 6, the curve does not cross at (2,0). So far
the sketch is...

The sketch now heads for the asymptote x = 1. It must go to plus infinity, since if it went to minus infinity, there
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would have to be another x intercept between I and 2 and there isn't. The power of (x - 1) is even. For an
asymptote, that means both ends are in the same location. Since one part is at plus infinity, so is the other. The
curve now heads through (0,8) toward (-1,0). It looks like...

Since the power of (x + 1) is odd, 7, there is a crossing at (-1,0). The sketch heads to minus infinity at x = -2.
Since the power of (x + 2) is odd and an odd power means one end at plus infinity and one at minus infinity,
one end is already at minus infinity, so that on the other side the curve goes to plus infinity. Remember, both
ends head for the horizontal asymptote y = 0, the x axis. The sketch can now be finished ....