Page 40 - Calc for the Clueless

Chapter 3
Curve Sketching Made Easy

The topic I think I can teach better than anyone else in the world is this one. The only question is whether I
could write it down. I think I did, and I think you'll really enjoy it!

Since we can spend an almost infinite amount of time on the topic, we will restrict our discussion to
polynomials and rational functions (polynomials over polynomials), except for a few examples at the end.

Terms and Special Notations

1. For curve-sketching purposes, we define an asymptote as a line to which the curve gets very close at the end
but never hits. All your life you have been told a curve cannot hit an asymptote. This is wrong. An asymptote is
a straight-line approximation to a curve near its end, that is, when x or y goes to plus or minus infinity. In the
middle of the curve, the curve is not a straight line and can hit the asymptote. The x axis is an asymptote
although the curve hits the axis five times. At the end of the curve, the curve gets close to the axis but does not
hit it.

2. | f(3) | = infinity. As x gets close to 3, f(x) gets very, very big (heading to plus infinity) or very, very small
(heading to minus infinity).

+
3. f(6 ). Substitute a number a little larger than 6, such as 6.01.
-
4. f(6 ). Substitute a number a little smaller than 6, such as 5.99.
Our first goal is to sketch, in under two minutes, curves like

Yes, not only is it possible, but almost all of my students do it and so will you!!!!!

Intercepts

x intercepts. Just like a straight line, x intercept means a point where y = 0. If we have a fraction, y = 0 means
the top of the fraction = 0.

Example 1—

4
y = 0 means the top is 0. ''Top is 0" means x = 0, 2x - 3 = 0, or x + 4 = 0. You ignore the exponents, since x = 0
means x = 0. x = 0, 3/2, and -4. The intercepts are (0,0), (1.5,0), and (-4,0).

Example 2—

Factor the top.