8.2-8.3: Graph Rational Functions II

8.2-8.3: Graph Rational Functions II
1.
2.
Objectives:
To find the slant
asymptotes of a
rational function
To graph rational
functions
Assignment:
• Graphing Rational
Functions II Worksheet
Objective 1
You will be able
asymptote of a
to find the slant
rational function
Rational Functions Mean Divide
Consider the rational function below.
2𝑥 2
𝑓 𝑥 = 2
𝑥 +1
We know that since d = n,
f has a horizontal
asymptote at y = 2.
2
−
2
𝑥2 + 1
2
2
𝑥 + 1 2𝑥 + 0𝑥 + 0
− 2𝑥 2
−2
Since a rational function is
−2
telling us to divide, let’s
do so.
Rational Functions Mean Divide
Consider the rational function below.
2𝑥 2
𝑓 𝑥 = 2
𝑥 +1
We know that since d = n,
f has a horizontal
asymptote at y = 2.
So 𝑓(𝑥) can be rewritten as:
2
𝑓 𝑥 =2− 2
Approaches 0 as x → ∞
𝑥 +1
And our graph is trying to look like y = 2 at large
values of x.
Rational Functions Mean Divide
Consider the rational function below.
2𝑥 3
𝑓 𝑥 = 2
𝑥 +1
We know that since d < n,
f has no horizontal
asymptotes.
2𝑥
−
2𝑥
𝑥2 + 1
2
3
2
𝑥 + 1 2𝑥 + 0𝑥 + 0𝑥 + 0
− 2𝑥 2
− 2𝑥
Since a rational function is
−2𝑥
telling us to divide, let’s
do so.
Rational Functions Mean Divide
Consider the rational function below.
2𝑥 3
𝑓 𝑥 = 2
𝑥 +1
We know that since d < n,
f has no horizontal
asymptotes.
So 𝑓(𝑥) can be rewritten as:
2𝑥
𝑓 𝑥 = 2𝑥 − 2
𝑥 +1
Approaches 0 as x → ∞
And our graph is trying to look like y = 2x at large
values of x. This is called a slant asymptote.
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 1
Can a rational function have both a slant
asymptote and a horizontal asymptote?
Exercise 2
Find all asymptotes of the rational function.
2𝑥 2 − 15𝑥 + 8
𝑓 𝑥 =
𝑥+3
Objective 2
You will be able to graph
rational functions
Graphing Algorithm
To graph a rational function:
1. Factor N(x) and D(x).
2. Find vertical asymptotes (where D(x) = 0) and plot as dashed
lines.
•
If a factor cancels, it is not an asymptote (A Hole)
3. Find horizontal asymptote (comparing d and n) and plot as a
dashed line.
4. Find slant asymptote (by long division w/o the remainder) and
plot as a dashed line.
5. Plot x- and y-intercepts.
•
If a factor cancels, it is not a zero (A Hole)
6. Use smooth curves to finish the graph.
More on Asymptotes
Vertical Asymptotes:
• Your graph can never cross one!
• If x = a is a vertical asymptote, then
interesting things happen really close to a:
– 𝑓(𝑥) could approach +∞ or −∞
– Think of vertical asymptotes as black holes
that attract values near a
More on Asymptotes
Vertical Asymptotes:
The end behavior around a vertical asymptote is
similar to that of polynomials:
V.A. at 𝑥 = 1 (multiplicity of 1)
V.A. at 𝑥 = 1 (multiplicity of 1)
More on Asymptotes
Vertical Asymptotes:
The end behavior around a vertical asymptote is
similar to that of polynomials:
V.A. at 𝑥 = 1 (multiplicity of 2)
V.A. at 𝑥 = 1 (multiplicity of 2)
More on Asymptotes
Horizontal Asymptotes:
• Your graph can cross
one!
• Attracts values
approaching +∞
or −∞
More on Asymptotes
Slant Asymptotes:
• Your graph can cross
one of these, too!
• Attracts values
approaching +∞
or −∞
Exercise 3
Graph:
𝑓 𝑥 =
1
𝑥+3
Exercise 4
Graph:
𝑓 𝑥 =
2𝑥 − 1
𝑥
Exercise 5
Graph:
𝑓 𝑥 =
3𝑥
𝑥2 + 𝑥 − 2
Exercise 6
Graph:
𝑓 𝑥 =
3𝑥
𝑥+2 𝑥−1
2
Exercise 7
Graph:
𝑓 𝑥 =
3𝑥
𝑥+2 2 𝑥−1
2
Exercise 8
Graph:
𝑥2 − 4
𝑓 𝑥 = 2
𝑥 + 4𝑥 + 4
Exercise 9
Graph:
𝑥2 + 𝑥 − 6
𝑓 𝑥 = 3
𝑥 + 3𝑥 2
Exercise 10
Graph:
𝑥2 − 𝑥
𝑓 𝑥 =
𝑥+1
Exercise 11
Graph:
𝑥2 − 𝑥 − 2
𝑓 𝑥 = 3
𝑥 − 2𝑥 2 − 5𝑥 + 6
Exercise 12
Graph:
𝑥3
𝑓 𝑥 = 2
2𝑥 − 8
8.2-8.3: Graph Rational Functions II
1.
2.
Objectives:
To find the slant
asymptotes of a
rational function
To graph rational
functions
Assignment
• Graphing Rational
Functions II
Worksheet
“Those gwafs are my favorite!”