Chapter 4

Day 2

What if the graph has an asymptote or two??

Find the first derivative. Put zeroes on a number line

Find the second derivative. Put zeroes on a number line

Lets look at the graph here:

x

yx 1

What do we do?The same general process holds.

1. Find vertical asymptotes and create “areas”2. Use f′(x) and determine max/min points in

each area. Plot these points3. Use f′′(x) to determine concavity in each

area. Plot these points. 4. If horizontal asymptotes exist, use them to

finish off the graph5. Plot x, y intercepts.

Use numerator AND denominator zeroes! Things can change around an

asymptote

Examples

2xf(x)

x 2

2

12. f(x)

x 4