Rational Functions and Their Graphs Section 3'5

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Rational Functions and Their GraphsSection 3.5

• JMerrill,2005
• Revised 08

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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 (CFOs)

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Introduction to Rational Functions

• What is a rational number?
• So just for grins, what is an irrational number?
• A rational function has the form

A number that can be expressed as a fraction
A number that cannot be expressed as a fraction
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Parent Function

• The parent function is
• The graph of the parent rational function looks
  like.
• The graph is not continuous and has asymptotes

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Transformations

• The parent function
• How does this move?

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Transformations

• The parent function
• How does this move?

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Transformations

• The parent function
• And what about this?

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Transformations

• The parent function
• How does this move?

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Transformations
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Domain
Find the domain of
Think what numbers can I put in for x????
Denominator cant equal 0 (it is undefined there)
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You Do Domain
Find the domain of
Denominator cant equal 0
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You Do Domain
Find the domain of
Denominator cant equal 0
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Vertical Asymptotes
At 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.
none
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Vertical Asymptotes
The figure below shows the graph of
The equation of the vertical asymptote is
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Vertical Asymptotes
Definition The line x a is a vertical
asymptote of the graph of f(x) if
as x approaches a either from the left or from
the right.
or
Look at the table of values for
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Vertical Asymptotes
As x approaches____ from the _______, f(x)
approaches _______.
As x approaches____ from the _______, f(x)
approaches _______.
-2
-2
right
left
Therefore, by definition, there is a vertical
asymptote at
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Vertical Asymptotes - 4
Describe what is happening to x and determine if
a vertical asymptote exists, given the following
information
Therefore, a vertical asymptote occurs at x -3.
As x approaches____ from the _______, f(x)
approaches _______.
As x approaches____ from the _______, f(x)
approaches _______.
-3
-3
left
right
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Vertical Asymptotes

• Set denominator 0 solve for x
• Substitute x-values into numerator. The values
  for which the numerator ? 0 are the vertical
  asymptotes

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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 its a vertical
  asymptote)

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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

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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!

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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.

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Example

• Function
• Solve denom.
• Factor and cancel
• Plug in -2

Hole is
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Asymptotes

• Some things to note
• Horizontal asymptotes describe the behavior at
  the ends of a function. They do not tell us
  anything about the functions 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?

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Horizontal Asymptotes
DefinitionThe line y b is a horizontal
asymptote if
as
or
Look at the table of values for
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Horizontal Asymptotes
0
0
y?_____ as x?________
y?____ as x?________
Therefore, by definition, there is a horizontal
asymptote at y 0.
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Examples
Horizontal Asymptote at y 0
Horizontal Asymptote at y 0
What similarities do you see between problems?
The degree of the denominator is larger than the
degree of the numerator.
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Examples
Horizontal Asymptote at
Horizontal Asymptote at y 2
What similarities do you see between problems?
The degree of the numerator is the same as the
degree or the denominator.
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Examples
No Horizontal Asymptote
No Horizontal Asymptote
What similarities do you see between problems?
The degree of the numerator is larger than the
degree of the denominator.
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Asymptotes Summary
1. 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 lt 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 gt d, then the
graph of f has ______ horizontal asymptote.
zeros
line y 0
no
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You Do
Find all vertical and horizontal asymptotes of
the following function
Vertical Asymptote x -1
Horizontal Asymptote y 2
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You Do Again
Find all vertical and horizontal asymptotes of
the following function
Vertical Asymptote none
Horizontal Asymptote y 0
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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.
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Example
Find all asymptotes.
Vertical
Horizontal
Slant
none
x 1
y x
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Example

• Find all asymptotes

Vertical asymptote at x 1
y x 1
n gt d by exactly one, so no horizontal asymptote,
but there is an oblique asymptote.
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Solving and Interpreting a Given Scenario
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The Average Cost of Producing a Wheelchair

• A company that manufactures wheelchairs has costs
  given by the function C(x) 400x
  500,000, where the x is the number of wheelchairs
  produced per month and C(x) is measured in
  dollars. The average cost per wheelchair for the
  company is given by

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Original C(x) 400x 500,000

• C(x) 400x 500,000
• x
• Find the interpret C(1000), C(10,000),
  C(100,000).
• C(1000) 900 the average cost of producing 1000
  wheelchairs per month is 900.

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C(x) 400x 500,000 x

• Find the interpret C(10,000)
• C(10,000) 450 the average cost of producing
  10,000 wheelchairs per month is 450.
• Find the interpret C(100,000)
• C(100,000) 405 the average cost of producing
  100,000 wheelchairs per month is 405.

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C(x) 400x 500,000 x

• What is the horizontal asymptote for the average
  cost function?
• Since n d (in degree) then y 400
• Describe what this represents for the company.

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C(x) 400x 500,000 x

• The horizontal asymptote means that the more
  wheelchairs produced per month, the closer the
  average cost comes to 400. Lower prices take
  place with higher production levels, posing
  potential problems for small businesses.