3-6: Curve Sketching
Objective:
1. To analyze and
sketch the graph of a
function
Assignment:
• P. 215-216: 1-6, 7-33
odd, 39-45 eoo, 49-52
Objective 1
You will be able to analyze and
sketch the graph of a function
Stuff You Need
When sketching the graph of a function, here
is the stuff you need:
Where 𝑓′ 𝑥 = 0
Where 𝑥 = 0
Domain and
Range
Where undefined
As 𝑥 → ∞
Where 𝑦 = 0
Intercepts, Holes,
Asymptotes, End
Behavior, and
Symmetry
Extrema,
Concavity, and
Points of Infection
Where 𝑓′′ 𝑥 = 0
Odd: 𝑓 −𝑥 = −𝑓 𝑥
Even: 𝑓 −𝑥 = 𝑓 𝑥
Exercise 1
Analyze and sketch the graph of 𝑓 𝑥 =
2 𝑥 2 −9
𝑥 2 −4
.
Exercise 2
Analyze and sketch the
graph of 𝑓 𝑥 =
𝑥 2 −2𝑥+4
.
𝑥−2
Slant Asymptotes
A rational function has a slant asymptote if
𝑛 =𝑑+1
– The degree of the numerator is one more than the
degree of the denominator
To find the equation of a slant asymptote, use
long division and forget about the remainder.
– At large values of 𝑥, the remainder approaches 0
anyway.
Exercise 3
Analyze and sketch the graph of 𝑓 𝑥 =
𝑥
.
𝑥 2 +2
Exercise 4
Analyze and sketch
the graph of
𝑓 𝑥 = 2𝑥 5/3 − 5𝑥 4/3 .
Exercise 5
Analyze and sketch the graph of
𝑓 𝑥 = 𝑥 4 − 12𝑥 3 + 48𝑥 2 − 64𝑥.
Exercise 6
Analyze and sketch the graph of 𝑓 𝑥 =
cos 𝑥
.
1+sin 𝑥
3-6: Curve Sketching
Objective:
1. To analyze and
sketch the graph of a
function
Assignment:
• P. 215-216: 1-6, 7-33
odd, 39-45 eoo, 49-52