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Rational Functions and Their Graphs
Why Should You Learn This? Rational functions are used to model and
solve many problems in the business world.
Some examples of real-world scenarios are:Average speed over a distance (traffic
engineers)Concentration of a mixture (chemist)Average sales over time (sales manager)Average costs over time (CFO’s)
Introduction to Rational Functions
What is a rational number?
So just for grins, what is an irrational number?
A rational function has the form ( )
( )( )
p xf x
q x
where p and q are polynomial functions
A number that can be expressed as a fraction:
A number that cannot be expressed as a fraction: , 2
5, 3, 4.5
2
What is Asymptote?
A line that a curve approaches as it heads towards infinity:
Types of Asymptote:
Horizontal Asymptotes
as x goes to infinity (or to -infinity) then the curve approaches some fixed constant value "b"
Types of Asymptote:
Vertical Asymptotes as x approaches
some constant value "c" (from the left or right) then the curve goes towards infinity (or -infinity)
Types of Asymptote:
Oblique Asymptotes as x goes to infinity
(or to -infinity) then the curve goes towards a line defined by y=mx+b (note: m is not zero as that would be horizontal).
The important point is that:
The distance between the curve and the asymptote tends to zero as they head to infinity
Got it!!!!
Parent Function The parent function is
The graph of the parent rational function looks like…………………….
The graph is not continuous and has asymptotes
1
x
Transformations
The parent function How does this move?
1
x 13
x
Transformations
The parent function How does this move?
1
x
1
( 3)x
Transformations
The parent function And what about this?
1
x
14
( 2)x
Transformations
The parent function
How does this move?
1
x
2
1
x
Transformations
2
1
x 2
12
x
2
14
( 3)x
2
1
( 3)x
Domain
Find the domain of 2x1f(x)
Denominator can’t equal 0 (it is undefined there)
2 0
2
x
x
Domain , 2 2,
Think: what numbers can I put in for x????
You Do: Domain
Find the domain of 2)1)(x(x
1-xf(x)
Denominator can’t equal 0
1 2 0
1, 2
x x
x
Domain , 2 2, 1 1,
You Do: Domain
Find the domain of 2
xf(x)x 1
Denominator can’t equal 02
2
1 0
1
x
x
Domain ,
Vertical AsymptotesAt the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below.
2x1f(x)
2x
2)1)(x(x1-xf(x)
1, 2x x
2
xf(x)x 1
none
Vertical Asymptotes
The figure below shows the graph of 2x1f(x)
The equation of the vertical asymptote is 2x
Vertical Asymptotes
Set denominator = 0; solve for x Substitute x-values into numerator. The
values for which the numerator ≠ 0 are the vertical asymptotes
Example
What is the domain? x ≠ 2 so
What is the vertical asymptote? x = 2 (Set denominator = 0, plug back into
numerator, if it ≠ 0, then it’s a vertical asymptote)
( , 2) (2, )
22 3 1( )
2
x xf x
x
You Do
Domain: x2 + x – 2 = 0 (x + 2)(x - 1) = 0, so x ≠ -2, 1
Vertical Asymptote: x2 + x – 2 = 0 (x + 2)(x - 1) = 0 Neither makes the numerator = 0, so x = -2, x = 1
( , 2) ( 2,1) (1, )
2
2
2 7 4( )
2
x xf x
x x
The graph of a rational function NEVER crosses a vertical asymptote. Why?
Look at the last example:
Since the domain is , and the vertical asymptotes are x = 2, -1, that means that if the function crosses the vertical asymptote, then for some y-value, x would have to equal 2 or -1, which would make the denominator = 0!
( , 1) ( 1,2) (2, )
2
2
2 7 4( )
2
x xf x
x x
Points of Discontinuity (Holes)
Set denominator = 0. Solve for x Substitute x-values into numerator. You
want to keep the x-values that make the numerator = 0 (a zero is a hole)
To find the y-coordinate that goes with that x: factor numerator and denominator, cancel like factors, substitute x-value in.
Example
Function:
Solve denom.
Factor and cancel
Plug in -2:
2
2
4( )
2 8
xf x
x x
2 2 8 0
( 4)( 2) 0
4, 2
x x
x x
x
( 2)( 2)
( 4)( 2)
x x
x x
( 2) 2 2 4 2
( 4) 2 4 6 3
x
x
22,
3 Hole is
Asymptotes
Some things to note: Horizontal asymptotes describe the behavior at the
ends of a function. They do not tell us anything about the function’s behavior for small values of x. Therefore, if a graph has a horizontal asymptote, it may cross the horizontal asymptote many times between its ends, but the graph must level off at one or both ends.
The graph of a rational function may or may not cross a horizontal asymptote.
The graph of a rational function NEVER crosses a vertical asymptote. Why?
Horizontal Asymptotes
Definition:The line y = b is a horizontal asymptote if f x b as x or x
Look at the table of values for f x 1
x 2
Horizontal Asymptotesx f(x)
1 .3333
10 .08333
100 .0098
1000 .0009
y→_____ as x→________
0
x f(x)
-1 1
-10 -0.125
-100 -0.0102
-1000 -0.001
y→____ as x→________
0
Therefore, by definition, there is a horizontal Therefore, by definition, there is a horizontal asymptote asymptote at y = 0.at y = 0.
Examples
f xx
( )
4
12f x
x
x( )
2
3 12
What similarities do you see between problems?
The degree of the denominator is larger than the degree of the numerator.
Horizontal Asymptote at y = 0
Horizontal Asymptote at y = 0
Examples
h xx
x( )
2 1
1 82x
15xg(x)
2
2
What similarities do you see between problems?
The degree of the numerator is the same as the degree or the denominator.
Horizontal Asymptote at y = 2
Horizontal Asymptote at 5
2y
Examples
13x
54x5x3xf(x)
23
2x
9xg(x)
2
What similarities do you see between problems?
The degree of the numerator is larger than the degree of the denominator.
No Horizontal Asymptote
No Horizontal Asymptote
Slant asymptotes Parabolic asymptotes
13x
54x5x3xf(x)
23
2x
9xg(x)
2
Slant or Oblique asymptotes only occur when the numerator of f(x) has a degree that is one higher than the degree of the denominator.
Parabolic asymptotes only occur when the numerator of f(x) is more than one higher than the degree of the denominator.
When you have this situation, simply divide the numerator by the When you have this situation, simply divide the numerator by the denominator, using polynomial long division or synthetic division. The denominator, using polynomial long division or synthetic division. The quotient (set equal to y) will be the oblique asymptote. Note that the quotient (set equal to y) will be the oblique asymptote. Note that the remainder is ignored.remainder is ignored.
Asymptotes: Summary1. The graph of f has vertical asymptotes at the _________ of q(x).
2. The graph of f has at most one horizontal asymptote, as follows:
a) If n < d, then the ____________ is a horizontal asymptote.
b) If n = d, then the line ____________ is a horizontal asymptote (leading coef. over leading coef.)
c) If n > d, then the graph of f has ______ horizontal asymptote.
zeros
line y = 0
no
ay
b
You DoFind all vertical and horizontal asymptotes of the following function
2 1
1
xf x
x
Vertical Asymptote: x = -1
Horizontal Asymptote: y = 2
You Do AgainFind all vertical and horizontal asymptotes of the following function
2
4
1f x
x
Vertical Asymptote: none
Horizontal Asymptote: y = 0
Oblique/Slant Asymptotes
The graph of a rational function has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator. Long division is used to find slant asymptotes.
The only time you have an oblique asymptote is when there is no horizontal asymptote. You cannot have both.
When doing long division, we do not care about the remainder.
ExampleFind all asymptotes.
2 2
1
x xf x
x
Vertical
x = 1
Horizontal
none
Slant
2
2
1 2
-2
x
x x x
x x
y = x
Example
Find all asymptotes: 2 2
( )1
xf x
x
Vertical asymptote at x = 1
n > d by exactly one, so no horizontal asymptote, but there is an oblique asymptote.
2
2
11 2
2
( 1)
1
-
xx x
x x
x
x
y = x + 1