Rational Functions

1
Chapter 7

• Rational Functions

2
Rational Function

• If a function is made up of the ratio of two
  polynomials then we say the function is rational.
  Examples

3
Graphs of Rational Functions

• What is the domain of the function given by

4
Do the Following

• Load the function f into your calculator and do a
  zoom decimal. Trace the graph from the left of 2
  and then from the right of 2. What appears to be
  happening?
• Explain what happens at x 2.

5
Vertical Asymptotes

• If a rational function is in reduced form and the
  value of a causes division by 0 (it is not in the
  domain of the function), then the vertical line x
  a is a vertical asymptote of the rational
  function and the function will not cross this
  line.

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• Find the vertical asymptote(s) for the following

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Reduced or Simplify

• We need to reduce our rational functions prior to
  finding the vertical asymptote. Recall

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Reduced or Simplify

• Also Recall

9

• Simplify the following rational expressions

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WARNING!

• Explain why the following statements are false

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Principle of Zero Products

• For any real numbers a and b

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Graphs of Rational Functions

• To find the x-intercepts of a rational function,
  we need only set the numerator equal to 0.
• To find the vertical asymptotes, we simplify the
  function and then find the values excluded from
  the domain. Reduced factors will make a hole in
  the graph.

13

• Graph the following and look at table. What are
  the equations of any vertical asymptotes?

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Multiplication and Division

• Section 7.2

15

• To multiply or divide rational expressions we
  need to recall how we did it with rational
  numbers

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• Simplify and Multiply the following

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Addition and Subtraction

• Section 7.3

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Addition of Rational Expressions

• To add or subtract when denominators are the
  same, add or subtract the numerators over the
  common denominator.

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Simplify
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Finding the LCD

• The Least Common Denominator (LCD) is the
  smallest possible expression that contains all of
  the factors of the denominators. Recall how to do
  it with fractions

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Finding the LCD

• The LCD is the least common multiple of the
  denominators. It should contain each factor the
  greatest number of times it occurs in any of the
  individual factorizations.

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Examples

• Find the LCD for the following sets of
  denominators

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Adding Unlike Denominators

• Section 7.4

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• Now that we can find an LCD we can add rational
  expressions with unlike denominators. Use the
  following steps
• (1) Find the LCD.
• (2) Rewrite each expression in the sum (or
  difference) with the LCD.
• (3) Add the numerators over the LCD.
• (4) See if the resulting expression simplifies.

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Examples

• Page 521 11, 20, 28, 36, 38, 48, 61

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Electrical Resistance

• If two resistors are put in a circuit in parallel
  (see picture) then their combine resistance can
  be modeled bySolve for R

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Section 7.5

• Complex Fractions

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How to Simplify a Complex Fraction

• Find the LCD of all expressions in the complex
  fraction
• Multiple by 1 in the form of LCD/LCD
• Simplify

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Examples
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Example
31
Section 7.6

• Rational Equations.

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How To Solve a Rational Equation

• Find the LCD of all of the fractions present
• Multiply both sides of the equation by the LCD
  (being careful to distribute)
• Solve the resulting equation
• Check your solution(s) in the original equation.

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Solve Both Graphical and Algebraic
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Asymptotes Again

• What are the vertical asymptotes of
• Does the graph show the same result?

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Section 7.7

• Applications of Rational Functions

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Recall the Steps

• Familiarize
• Translate
• Solve
• Check
• State

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Modeling Work

• If a amount of time for A aloneb amount of
  time for B alonet the time for A and B
  together
• Then

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Example

• James can fill a ditch in 9hrs. Danika can fill
  it in 5 hrs. How long will it take them working
  together?

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Example

• Oscar takes 3 hr longer to paint a floor than
  Sasha. When they work together it takes them 2
  hrs. How long would each take working alone?

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Motion

• Recall the relation D rtwhere D is the
  distance, r is the rate, and t is the time. For
  example, if you traveled 276 miles in 4 hours,
  what would your average rate be?

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Example

• The current in the Mad River moves at a rate of 4
  mph in the summer. If Krista paddles her kayak 6
  miles upstream in the same time it takes her to
  paddle 12 miles down stream, what is her speed in
  still water?

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Similar Triangles

• Proportions show up in geometry all the time. One
  classic example is similar triangles.

B
S
a
c
t
r
R
T
A
C
b
s
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Example

• For the pair of given similar triangles find the
  value of the indicated side.

B
S
7
4
R
T
A
C
b
6
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Ratios in Populations

• A commonly used technique to estimate the size of
  a population is the mark recapture procedure. A
  game warden tags 318 deer in a national forest
  and at the end of the hunting season 168 deer are
  caught 56 of them have tags. Estimate the number
  of deer in the forest.

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Formulas

• There are many places in science and mathematics
  where a formula is used to describe the
  relationship between values. Geometry is full of
  cool formulas like P 2L 2W for the perimeter
  of a rectangle.
• Rearranging a formula is a valuable skill.

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Example from Finance

• The amount of money in a savings account is given
  by

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Examples from Geometry
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Variation

• Direct Variation When a situation gives rise to
  a linear function of the formwhere k is a
  nonzero constant, we say that y varies directly
  with x.

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Variation

• Inverse Variation When a situation gives rise to
  a rational function of the formwhere k is a
  nonzero constant, we say that y varies inversely
  with x.

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Examples

• Page 566 33. 37, 38, 45, 51, 55