Math 4 lecture on Graphing Rational Functions

Graphing Rational Functions

Mathematics 4

July 26, 2011

Mathematics 4 () Graphing Rational Functions July 26, 2011 1 / 61

Review of Previous Concepts on Rational Functions

Rational Functions



Rational Functions
p(x)
A rational function can be expressed as f (x) = where p(x)
q(x)
and q(x) are polynomial functions.



Rational Functions
p(x)
q(x)
The domain of f (x) is R excluding the values where the denominator
q(x) is zero.



Rational Functions
p(x)
q(x)
q(x) is zero.
The zeros of the numerator p(x) are the zeros of the rational function
f (x).



Rational Functions
p(x)
q(x)
q(x) is zero.
The zeros of the numerator p(x) are the zeros of the rational function
f (x).
If f (x) is in lowest terms, the zeros of the denominator q(x)
determine the vertical asymptotes of the function.


The Horizontal Asymptote

Horizontal Asymptotes



The horizontal asymptote is an imaginary line that a rational
function approaches as x → −∞ and x → ∞.



The horizontal asymptote is an imaginary line that a rational
function approaches as x → −∞ and x → ∞.
The graph of a rational function CAN cross its horizontal asymptotes.



Consider the function below:
3x + 5
f (x) =
x−5
f (−100) =



3x + 5
f (x) =
x−5
f (−100) = 2.809
f (−1000) =



3x + 5
f (x) =
x−5
f (−100) = 2.809
f (−1000) = 2.98009
f (−10000) =



3x + 5
f (x) =
x−5
f (−100) = 2.809 f (100) =
f (−1000) = 2.98009
f (−10000) = 2.998



3x + 5
f (x) =
x−5
f (−100) = 2.809 f (100) = 3.2105
f (−1000) = 2.98009 f (1000) =
f (−10000) = 2.998



3x + 5
f (x) =
x−5
f (−100) = 2.809 f (100) = 3.2105
f (−1000) = 2.98009 f (1000) = 3.0201
f (−10000) = 2.998 f (10000) =



3x + 5
f (x) =
x−5
f (−100) = 2.809 f (100) = 3.2105
f (−1000) = 2.98009 f (1000) = 3.0201
f (−10000) = 2.998 f (10000) = 3.002



3x + 5
f (x) =
x−5
f (−100) = 2.809 f (100) = 3.2105
f (−1000) = 2.98009 f (1000) = 3.0201
f (−10000) = 2.998 f (10000) = 3.002

As x → −∞ and as x → ∞, the value of f (x) becomes closer to 3.



3x + 5
f (x) =
x−5
f (−100) = 2.809 f (100) = 3.2105
f (−1000) = 2.98009 f (1000) = 3.0201
f (−10000) = 2.998 f (10000) = 3.002

As x → −∞ and as x → ∞, the value of f (x) becomes closer to 3.

The horizontal asymptote is the line y = 3.



3x3 + 4x2 + 7x + 2
Consider the function f (x) =
4x3 − 5x2 + 3x + 1



3x3 + 4x2 + 7x + 2
4x3 − 5x2 + 3x + 1

Let’s analyze the function as x → −∞: (please refer to the blackboard)



3x3 + 4x2 + 7x + 2
4x3 − 5x2 + 3x + 1


Now analyze the function as x → ∞: (please refer to the blackboard)



3x3 + 4x2 + 7x + 2
4x3 − 5x2 + 3x + 1



The horizontal asymptote is y = 3 .
4



degree of p(x) = degree of q(x)
If the degree of the numerator p(x) is equal to the degree of the
denominator q(x), the horizontal asymptote is
an
y=
bn
where an is the leading coeﬃcient of p(x) and q(x) is the leading
coeﬃcient of q(x).



5x + 2
3x3 − x2 − 8x − 2



5x + 2
3x3 − x2 − 8x − 2



The horizontal asymptote is y = 0.



degree of p(x) < degree of q(x)
If the degree of the numerator p(x) is less than the degree of the
denominator q(x), the horizontal asymptote is

y=0



Summary
an xn + ... + a1 x + a0
The graph of f (x) = has:
bm xm + ... + b1 x + b0

its HA at the x-axis n<m



Summary
an xn + ... + a1 x + a0
bm xm + ... + b1 x + b0

an
its HA at y = n=m
bm



Summary
an xn + ... + a1 x + a0
bm xm + ... + b1 x + b0

an
its HA at y = n=m
bm
no horizontal asymptote n>m



Graphing
The following information is needed to graph a rational function f (x):



Graphing
1 Domain of f (x)



Graphing
1 Domain of f (x)
2 Zeros of f (x)



Graphing
1 Domain of f (x)
2 Zeros of f (x)
3 The vertical asymptotes (if any)



Graphing
1 Domain of f (x)
2 Zeros of f (x)
4 The horizontal asymptote (if any)



Graphing
1 Domain of f (x)
2 Zeros of f (x)
5 The y-intercept



Graphing
1 Domain of f (x)
2 Zeros of f (x)
5 The y-intercept
6 The table of signs



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Domain:



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Domain: x=1



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Zeros:



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Zeros: x = −1



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Vertical asymptotes:



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Vertical asymptotes: x=1



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Horizontal asymptote:



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

Horizontal asymptote: y=1



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

y-intercept:



x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

y-intercept: −1


x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

−1 1
x+1 - 0 + + +
Table of signs:
x−1 - - - 0 +


x+1
Example: f (x) =
x−1
y
5

4

3

2

1
x

−5 −4 −3 −2 −1 1 2 3 4
−1

−2

−3

−1 1
x+1 - 0 + + +
Table of signs:
x−1 - - - 0 +
f (x) + 0 − VA +



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

Domain:



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

Domain: x = −1, 3



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

Zeros



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

Zeros −3, 1



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

y-intercept



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

y-intercept: −1



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

vertical asymptote:



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

vertical asymptote: x = −1, x = 3



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

horizontal asymptote:



(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

horizontal asymptote: y = −1


(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

−3 −1 1 3
x+3 - 0 + + + + + + +
x−1 - - - - - 0 + + +
Table of signs:
x+1 - - - 0 + + + + +
3−x + + + + + + + 0 -


(x + 3)(x − 1)
Example: f (x) =
(x + 1)(3 − x)
5
y
4

3

2

1
x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1

−2

−3

−4

−5

−3 −1 1 3
x+3 - 0 + + + + + + +
x−1 - - - - - 0 + + +
Table of signs:
x+1 - - - 0 + + + + +
3−x + + + + + + + 0 -
f (x) − 0 + VA − 0 + VA −

Assignment: Homework 12 due tomorrow.


Graphing Rational Functions with Oblique Asymptotes

Oblique Asymptote
n
A rational function f (x) = bmx m+...+a11x+a0 has an oblique asymptote if it
an
x +...+b x+b0
is in lowest terms and n = m + 1.



Oblique Asymptote
n
A rational function f (x) = bmx m+...+a11x+a0 has an oblique asymptote if it
an
x +...+b x+b0
is in lowest terms and n = m + 1.

Functions with oblique asymptotes
x2 − 2x + 1
f (x) =
x
x 2 + 6x + 8
f (x) =
x+3
x3 + 7x2 − 20x − 96
f (x) =
x2 − 10x + 16



Oblique Asymptote
The oblique asymptote is a line of the form y = mx + b.



Oblique Asymptote
The oblique asymptote is a line of the form y = mx + b.

Finding the oblique asymptote
Take the polynomial part of the long division of the rational function.



Oblique Asymptote
x2 −2x+1
Find the oblique asymptote of f (x) = x :



Oblique Asymptote
x2 −2x+1
Find the oblique asymptote of f (x) = x :

x−2
x x2 − 2x + 1
− x2
− 2x
2x

The oblique asymptote is y = x − 2.



Oblique Asymptote
x2 +6x+8
Find the oblique asymptote of f (x) = x+3 :



Oblique Asymptote
x2 +6x+8
Find the oblique asymptote of f (x) = x+3 :

x+3
x+3 x2
+ 6x + 8
− x2 − 3x
3x + 8
− 3x − 9
−1
The oblique asymptote is y = x + 3.



Oblique Asymptote
x3 +7x2 −20x−96
Find the oblique asymptote of f (x) = x2 −10x+16
:



Oblique Asymptote
x3 +7x2 −20x−96
Find the oblique asymptote of f (x) = x2 −10x+16
:

x + 17
x2 − 10x + 16 x3 + 7x2
− 20x − 96
− x3 + 10x2 − 16x
17x2 − 36x − 96
− 17x2 + 170x − 272
134x − 368
The oblique asymptote is y = x + 17.



Oblique Asymptote
3x3 −2x2 +4x−3
Find the oblique asymptote of f (x) = x2 +3x+3
:



Oblique Asymptote
3x3 −2x2 +4x−3
Find the oblique asymptote of f (x) = x2 +3x+3
:

3x − 11
x2 + 3x + 3 3x3− 2x2
+ 4x − 3
− 3x3 − 9x2 − 9x
− 11x2 − 5x − 3
11x2 + 33x + 33
28x + 30
The oblique asymptote is y = 3x − 11.



x2 +6x+8
Example: f (x) = x+3
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

Domain:



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

Domain: x = −3



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

Zeros:



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

Zeros: −4, −2



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

y-intercept:



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

8
y-intercept: 3



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

vertical asymptotes:



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

vertical asymptotes: x = −3



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

oblique asymptote:



x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

oblique asymptote: y =x+3


x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

−4 −3 −2
x+2 - - - - - 0
Table of signs: x+4 - 0
x+3 - - - 0


x2 +6x+8
y

4

3

2

1

x

−7 −6 −5 −4 −3 −2 −1 1 2

−1

−2

−3

−4

−4 −3 −2
x+2 - - - - - 0
x+3 - - - 0
f (x) − 0 + VA − 0 +


Math 4 lecture on Graphing Rational Functions

Math 4 lecture on Graphing Rational Functions

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