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