8.2-8.3: Graphing Rational Functions I

8.2-8.3: Graphing Rational Functions I
1.
2.
Objectives:
To find the domain
of rational functions
To find the vertical
and horizontal
asymptotes of
rational functions
and their zeros
Assignment:
• Graphing Rational
Functions I Worksheet
• Extending Rational
Functions Worksheet:
M3
Warm-Up
Why is division by zero undefined?
To answer the question, evaluate 𝑓 𝑥 = 1
for values of 𝑥 between 1 and 0, getting
closer and closer to zero.
Then evaluate 𝑓 𝑥 = 1
between −1 and 0.
𝑥
for values of 𝑥
𝑥
Warm-Up
Why is division by zero undefined?
x
1
.1
.01
.001
.0001
f(x) = 1/x
1
10
100
1000
10000
 
As x → 0+ (approaches 0 from the right), f (x)
increases without bound (approaches
positive infinity).
Warm-Up
Why is division by zero undefined?
x
−1
−.1
−.01
−.001
−.0001
f(x) = 1/x
−1
−10
−100
−1000
−10000
 
As x → 0− (approaches 0 from the left), f (x)
decreases without bound (approaches
negative infinity).
Warm-Up
Why is division by zero undefined?
Graphically, you can
see f (x) approach
+∞ from the right and
−∞ from the left. For
this graph x = 0 is a
vertical asymptote.
Also, y = 0 is a
horizontal
asymptote.
Warm-Up
Why is division by zero undefined?
This means the
graph is not
continuous at
x = 0. And so the
function is not
defined at x = 0.
Thus, you cannot
divide by zero.
Warm-Up
Why is division by zero undefined?
That, plus you
don’t want to
create a hole in
space and
time.
Warm-Up
Why is division by zero undefined?
The graph of f (x) = 1/x
is the parent function
of simple rational
functions.
The graph is called a
hyperbola where each
arm is called a branch
Warm-Up
Why is division by zero undefined?
The graph of f (x) = 1/x
is the parent function
of simple rational
functions.
You could SRT
transformations on
y
a
k
xh
Exercise 1
Let f (x) = 2x + 5 and g(x) = 3x − 7. Find
𝑓
𝑔 (𝑥) and its domain.
Rational Functions
A rational function can be written as
𝑁(𝑥)
𝑓 𝑥 =
𝐷(𝑥)
where N(x) and D(x) are polynomials,
and D(x) ≠ 0.
Domain of Rational Functions
In general, the domain of a rational function
is all real numbers except the x-values that
make D(x) = 0.
𝑁(𝑥)
𝑓 𝑥 =
𝐷(𝑥)
In fact, all the interesting bits of the graphs of
rational functions occur around these
excluded x-values.
Domain of Rational Functions
To find the domain of a rational function,
follow these 3 easy steps:
Set the
1 ≠
bottom
0.
Solve the
“inequation”
for x as2if you
were solving a
real equation.
These are
the x-values
that must be
3
excluded
from the
domain.
 x : x  a, b
Exercise 2
Find the domain of each function below.
1. 𝑓 𝑥 =
9𝑥 2 −9𝑥−10
2𝑥+8
2. 𝑓 𝑥 =
2𝑥+8
9𝑥 2 −9𝑥−10
Asymptotes
Vertical Asymptote:
The line x = a is a vertical asymptote of the
graph of f (x) if f (x) →+∞ or f (x) →−∞ as x → a.
Asymptotes
Horizontal Asymptote:
The line y = b is a horizontal asymptote of the
graph of f (x) if f (x) → b as x →+∞ or x →−∞ .
Exercise 3
Fact: The graphs of rational functions have
asymptotes.
Query: Can the graphs of rational functions
be continuous?
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Vertical Asymptotes:
Vertical
asymptotes
happen where
D(x) = 0.
These were the
values excluded
from the domain!
Easy!
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
In other words, the 𝑥-axis is a
horizontal asymptote when the
degree of the denominator is
greater than the numerator.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
This happens because the
denominator is increasing faster
than the numerator. Thus the
function approaches zero.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
If d = n, then
𝑦 = 𝑎𝑏𝑛𝑛 is a
horizontal
asymptote.
In other words,
when the degrees
are equal, the
function approaches
the ratio of the
leading coefficients.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
If d = n, then
𝑦 = 𝑎𝑏𝑛𝑛 is a
horizontal
asymptote.
This happens
because the
numerator and
denominator are
increasing/decreasing
at roughly the same
rate.
Finding Asymptotes
Let f be the rational function given below:
𝑁(𝑥) 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑓 𝑥 =
=
𝐷(𝑥) 𝑏𝑑 𝑥 𝑑 + 𝑏𝑑−1 𝑥 𝑑−1 + ⋯ + 𝑏1 𝑥 + 𝑏0
Horizontal Asymptotes:
If d > n, then
y = 0 is a
horizontal
asymptote.
If d = n, then
𝑦 = 𝑎𝑏𝑛𝑛 is a
horizontal
asymptote.
If d < n, then
there are no
horizontal
asymptotes.
This happens because the
numerator is increasing
faster than the
denominator.
Horizontal Asymptotes I
You could always memorize the rules for finding
horizontal asymptotes, or you could just try to
understand what’s happening mathematically:
1. When d > n:
𝑓 𝑥 =
𝑓 100 =
𝑥
𝑥2 + 1
Denominator increases faster than numerator.
100
100
≈ .009999
=
10001
10000 + 1
Approaches 0
as 𝑥 gets bigger
So y = 0 is a horizontal asymptote.
Horizontal Asymptotes II
You could always memorize the rules for finding
horizontal asymptotes, or you could just try to
understand what’s happening mathematically:
2. When d = n:
2𝑥 2
𝑓 𝑥 = 2
Both increase at roughly the same rate.
𝑥 +1
20000
20000
≈ 1.99998
=
𝑓 100 =
Approaches 2
10001
10000 + 1
as 𝑥 gets bigger
So y = 2 is a horizontal asymptote.
Horizontal Asymptotes III
You could always memorize the rules for finding
horizontal asymptotes, or you could just try to
understand what’s happening mathematically:
3. When d < n:
𝑥2 + 1
Numerator increases faster than denominator.
𝑓 𝑥 =
𝑥
Increases
10001
10000 + 1
without bound
≈ 100.01
=
𝑓 100 =
100
100
as 𝑥 gets bigger
So there are no horizontal asymptotes.
Exercise 4
Identify all vertical and horizontal asymptotes. Then
find the zeros of each function.
1. 𝑓 𝑥 =
9𝑥 2 −9𝑥−10
2𝑥+8
2. 𝑓 𝑥 =
2𝑥+8
9𝑥 2 −9𝑥−10
Exercise 5
Identify all vertical and horizontal asymptotes. Then
find the zeros of each function.
1. 𝑓 𝑥 =
6𝑥 2 +9𝑥−15
3𝑥 2 −21𝑥
2. 𝑓 𝑥 =
3𝑥 2 −21𝑥
6𝑥 2 +9𝑥−15
Exercise 6
Identify all vertical and horizontal asymptotes. Then
find the zeros of each function.
1. 𝑓 𝑥 =
2𝑥 2 −7𝑥−15
2𝑥 2 −5𝑥−12
2. 𝑓 𝑥 =
𝑥 2 +𝑥−6
𝑥 3 +3𝑥 2
Exercise 6
Identify all vertical and horizontal asymptotes. Then
find the zeros of each function.
1. 𝑓 𝑥 =
2𝑥 2 −7𝑥−15
2𝑥 2 −5𝑥−12
A hole
Horizontal
Asymptote
2. 𝑓 𝑥 =
Vertical
Asymptote
𝑥 2 +𝑥−6
𝑥 3 +3𝑥 2
Horizontal
Asymptote
Vertical
Asymptote
Zero
Zero
A hole
Factors That Cancel
As the previous Exercise demonstrated,
sometimes a factor in the numerator will
cancel with a factor in the denominator.
When this happens:
1. There is no vertical asymptote at the
canceled factor.
2. There is A Hole in the graph at the
canceled factor, and it is not a zero.
8.2-8.3: Graphing Rational Functions I
1.
2.
Objectives:
To find the domain
of rational functions
To find the vertical
and horizontal
asymptotes of
rational functions
and their zeros
Assignment
• Graphing Rational
Functions I
Worksheet
• Extending Rational
Functions
Worksheet: M3
“Give me an asymptotic high five!”