1. Section 3.6 A Summary of Curve Sketching
AP Calculus BC

2. Warm-up:

3. Warm-up:

4. Warm-up -- Solution:

5. Learning Objective:
1.) Analyze and sketch the graph of a function.

6. Tools/concepts for curve sketching
All the different components of a graph that would help us determine its behavior are listed below. Some of these will be names of specific points on a graph, while some will be graph qualities or characteristics.
x- and y-intercepts vertical & horizontal asymptotes
relative and absolute max/min domain & range
increasing & decreasing intervals continuity
points of inflection differentiability
concavity symmetry (even/odd functions)

7. Steps for Curve Sketching:
Use Algebra –
Find Asymptotes (use this to determine the domain/range)
Find x-intercepts
Find y-intercepts
Use Calculus
Find 𝑓 ′ (𝑥) and critical value(s)
Find where 𝑓 𝑥 is increasing/decreasing
Find 𝑓 ′′ (𝑥) and point(s) of inflection(s)
Find where 𝑓 𝑥 is concave up/concave down

8. Example 1: Analyze and sketch the graph of: 𝑓 𝑥 = 2( 𝑥 2 −9) 𝑥 2 −4
Example on p. 207

9. Example 2: Analyze and sketch the graph of: 𝑓 𝑥 = 𝑥 𝑥 2 +2
Example on p. 207

10. Example 3: Analyze and sketch the graph of: 𝑓 𝑥 =2 𝑥 5 3 −5 𝑥 4 3
Example on p. 207

11. Example 4:
Analyze and sketch the graph of: 𝑓 𝑥 = 𝑥 4 −12 𝑥 𝑥 2 −64𝑥
**Note: A polynomial function of even degree must have at least one relative extrema.
Example on p. 207

12. Example 5: Analyze and sketch the graph of: 𝑓 𝑥 = cos⁡(𝑥) 1+sin⁡(𝑥) .
Example on p. 207

13. Homework:
pg : #1-4 all, 5-23 odd, 37, 38, 49, 51