Rational Functions

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Chapter 9

• Rational Functions

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In this chapter you should

• Learn to use inverse variation and the graphs of
  inverse variations to solve real-world problems.
• Learn to identify properties of rational
  functions.
• Learn to simplify rational expressions and to
  solve rational equations.

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2-3 Direct Variation

• What youll learn
• To write and interpret direct variation equations

• 1.05 Model and solve problems using direct,
  inverse, combined and joint variation.

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• This is a graph of direct variation.  If the
  value of x is increased, then y increases as
  well.Both variables change in the same manner. 
  If x decreases, so does the value of y.  We say
  that y varies directly as the value of x. 

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Definition Y varies directly as x means that y
kx where k is the constant of
variation. (see any similarities to y mx
b?) Another way of writing this is k
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Example 1a Identifying Direct Variation from a
Table

• For each function, determine whether y
  varies directly with x. If so, find the constant
  of variation and write the equation.

x y
1 4
2 7
5 16
x y
2 8
3 12
5 20
k _______
k _______
Equation _________________
Equation _________________
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Example 1b Identifying Direct Variation from a
Table

• For each function, determine whether y
  varies directly with x. If so, find the constant
  of variation and write the equation.

x y
-1 -2
3 4
6 7
x y
-6 -2
3 1
12 4
k _______
k _______
Equation _________________
Equation _________________
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Example 4a Using a Proportion

• Suppose y varies directly with x, and x 27 when
  y -51. Find x when y -17.

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Example 4b Using a Proportion

• Suppose y varies directly with x, and x 3 when
  y 4. Find y when x 6.

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Example 4c Using a Proportion

• Suppose y varies directly with x, and x -3 when
  y 10. Find x when y 2.

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9-1 Inverse Variation

• What youll learn
• To use inverse variation
• To use combined variation

• 1.05 Model and solve problems using direct,
  inverse, combined and joint variation.

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• In an inverse variation, the values of the two
  variables change in an opposite manner - as one
  value increases, the other decreases.  Inverse
  variation  when one variable increases,the
  other variable decreases. 

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Inverse Variation
When two quantities vary inversely, one quantity
increases as the other decreases, and vice versa.
Generalizing, we obtain the following statement.
An inverse variation between 2 variables, y and
x, is a relationship that is expressed as
            where the variable k is called the
constant of proportionality. As with the direct
variation problems, the k value needs to be
found using the first set of data.
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Example 2a Identifying Direct and Inverse
Variation

• Is the relationship between the variables in
  each table a direct variation, an inverse
  variation, or neither? Write functions to model
  the direct and inverse variations.

x 0.5 2 6
y 1.5 6 18
x 0.2 0.6 1.2
y 12 4 2
x 1 2 3
y 2 1 0.5
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Example 2a Identifying Direct and Inverse
Variation

• Is the relationship between the variables in
  each table a direct variation, an inverse
  variation, or neither? Write functions to model
  the direct and inverse variations.

x 0.8 0.6 0.4
y 0.9 1.2 1.8
x 2 4 6
y 3.2 1.6 1.1
x 1.2 1.4 1.6
y 18 21 24
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Example 3 Real World Connection

• Zoology. Heart rates and life spans of most
  mammals are inversely related. Us the data to
  write a function that models this inverse
  variation. Use your function to estimate the
  average life span of a cat with a heart rate of
  126 beats / min.

Mammal Heart Rate (beats per min) Life Span (min)
Mouse 634 1,576,800
Rabbit 158 6,307,200
Lion 76 13,140,000
Horse 63 15,768,000
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• A combined variation combines direct and
  inverse variation in more complicated
  relationships.

Combined Variation Equation Form
y varies directly with the square of x y kx2
y varies inversely with the cube of x y
z varies jointly with x and y. z kxy
z varies jointly with x and y and inversely with w. z
z varies directly with x and inversely with the product of w and y. z
k x3
kxy w
kx wy
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Example 5a Finding a Formula

• The volume of a regular tetrahedron varies
  directly as the cube of the length of an edge.
  The volume of a regular tetrahedron with edge
  length 3 is .
• Find the formula for the volume of a regular
  tetrahedron.

e
e
9 v 2 4
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Example 5b Finding a Formula

• The volume of a square pyramid with congruent
  edges varies directly as the cube of the length
  of an edge. The volume of a square pyramid with
  edge length 4 is .
• Find the formula for the volume of a square
  pyramid with congruent edges.

e
e
e
32 v 2 3
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9-3 Rational Functions and Their Graphs

• What youll learn
• To identify properties of rational functions
• To graph rational functions

• 2.05 Use rational equations to model and solve
  problems justify results.
• Solve using tables, graphs, and algebraic
  properties.
• Interpret the constants and coefficients in the
  context of the problem.
• Identify the asymptotes and intercepts
  graphically and algebraically.

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• Definition Rational Function
• A rational function f(x) is a function that
  can be written as
• where P(x) and Q(x) are polynomial functions
  and Q(x) ? 0.

P(x) Q(x)
f(x)
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Examples of Rational Functions
-2x x2 1
y

• In this graph, there is no value of x that
  makes the denominator 0. The graph is continuous
  because it has no jumps, breaks, or holes in it.
  It can be drawn with a pencil that never leaves
  the paper.

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Examples of Rational Functions
1 x2 - 16
y

• In this graph, x cannot be 4 or -4 because then
  the denominator would equal 0.

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Examples of Rational Functions
(x2)(x-1) x - 1
y

• In this graph, x cannot equal 1 or the
  denominator would equal 0.

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Point of Discontinuity

• A function is said to have a point of
  discontinuity at x a or the graph of the
  function has a hole at x a, if the original
  function is undefined for x a, whereas the
  related rational expression of the function in
  simplest form is defined for x a. 

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Example of Point of Discontinuity

• Consider a function .
• This function is undefined for x 2. But the
  simplified rational expression of this function,
  x 3 which is obtained by canceling (x - 2)
  both in the numerator and the denominator is
  defined at x 2. Thus we can say that the
  function f(x) has a point of discontinuity at x
  2.

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Example 1b Finding Points of Discontinuity
1 x2 - 16
x2 - 1 x2 3
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Vertical Asymptotes

• An asymptote is a line that the curve approaches
  but does not cross. The equations of the vertical
  asymptotes can be found by finding the roots of
  q(x). Completely ignore the numerator when
  looking for vertical asymptotes, only the
  denominator matters.
• If you can write it in factored form, then you
  can tell whether the graph will be asymptotic in
  the same direction or in different directions by
  whether the multiplicity is even or odd.
• Asymptotic in the same direction means that the
  curve will go up or down on both the left and
  right sides of the vertical asymptote. Asymptotic
  in different directions means that the one side
  of the curve will go down and the other side of
  the curve will go up at the vertical asymptote.

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Example 1a Finding Points of Discontinuity
1 x2 2x 1
-x 1 x2 1
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Example 2a Finding Vertical Asymptotes
(x 2) (x 1) x - 2
x 1 (x 2)(x 3)
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Example 2b Finding Vertical Asymptotes
x 2 (x - 1)(x 3)
(x 3)(x 4) (x 3)(x 3)(x4)
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Horizontal Asymptotes

• The graph of a rational function has at most one
  HA.
• The graph of a rational function has a HA at y0
  if the degree of the denominator is greater than
  the degree of the numerator .
• If the degrees of the numerator and the
  denominator are , then the graph has a HA at y
  , a is the coefficient of the term of the
  highest degree in the numerator and b is the
  coefficient of the term of the highest degree in
  the denominator.
• If the degree of the numerator is greater than
  the degree of the denominator, then the graph has
  no HA

a b
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Example 4a Sketching Graphs of HA
x 2 (x3)(x-4)

• y

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Example 4b Sketching Graphs of HA
x 3 (x-1)(x-5)

• y

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Example 5 Real World Connection

1. Write a function for the average cost of a
   salable disc. Graph the function.
2. What is the average cost if 2000 discs are
   produced? If 12,800 discs are produced?

• The CD-ROMs for a computer game can be
  manufactured for .25 each. The development cost
  is 124,000. The first 100 discs are samples and
  will not be sold.

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9-4 Rational Expressions

• What youll learn
• To simplify rational expression
• To multiply and divide rational expressions

• 1.03 Operate with algebraic expressions
  (polynomial, rational, complex fractions) to
  solve problems.

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• A rational expression is in its simplest form
  when its numerator and denominator are
  polynomials that have no common divisors.

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Example 1a Simplifying Rational Expressions

• -27x3y
• 9x4y

x2 10x 25 x2 9x 20
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Example 1b Simplifying Rational Expressions

• 2x2 3x - 2
• x2 5x 6

-6 3x x2 - 6x 8
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Example 2 Real World Connection

• Architecture One factor in designing a
  structure is the need to maximize the volume
  (space for working) for a given surface area
  (material needed for construction). Compare the
  ratio of the volume to surface area of a cylinder
  with radius r and height r to a cylinder with
  radius r and height 2r.

SA 2?rh 2?r2
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Multiplying Rational Expressions

• Simply Put The rule for multiplying algebraic
  fractions is the same as the rule for multiplying
  numerical fractions.
• Multiply the tops (numerators) AND multiply the
  bottoms (denominators). 

If possible, reduce (cancel) BEFORE you multiply
the tops and bottoms!(It's easier than
simplifying at the end!)
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Example 3a Multiplying Rational Expressions
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Example 3b Multiplying Rational Expressions
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Dividing Rational Expressions

• Simply PutThe rule for dividing algebraic
  fractions is the same as the rule for dividing
  numerical fractions.
• Change the division sign to multiplication,
  flip the 2nd fraction ONLY, and then follow the
  steps for "multiplying rational expressions".

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Example 4a Dividing Rational Expressions
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Example 4b Dividing Rational Expressions
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9-5 Adding and Subtracting Rational
Expressions

• What youll learn
• To add and subtract rational expressions
• To simplify complex fractions

• 1.03 Operate with algebraic expressions
  (polynomial, rational, complex fractions) to
  solve problems.

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The Basic RULE for Adding and Subtracting
Fractions

• Get a Common Denominator! (the smallest number
  that both denominators can divide into without
  remainders.) 
• With each fraction, whatever is multiplied times
  the bottom must ALSO be multiplied times the
  top.
• Do not add the common denominators.  Add only the
  numerators (tops).

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Adding and Subtracting Fractions with Like
Denominators

• 2 5
• 3 3

4 3 7 7

-
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Adding Expressions with Like Denominators

• 2 5
• x 3 x 3

y y 3 y 5 y
5


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Subtracting Expressions with Like Denominators

• 2n 1 3n 4
• 2n 5n 3 2n 5n 3

4 5 t 2 t
2
-
-
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Example 3a Adding Rational Expressions with
Unlike Denominators
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Example 3b Adding Rational Expressions with
Unlike Denominators
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Example 4a Subtracting Rational Expressions with
Unlike Denominators
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More Examples
1 x2 5x 4
5x 3x 3

7y 5y2 - 125
4 3y 15
-
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•    
• A complex fraction is a fraction in which the
  numerator, denominator, or both, also contain
  fractions.
• If the complex fraction contains a variable, it
  is called a complex rational expression.
• Simplify complex fractions by multiplying by a
  common denominator.

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Example 5a Simplifying Complex Fractions

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Example 5b Simplifying Complex Fractions

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9-6 Solving Rational Equations

• What youll learn
• To solve rational expressions
• To use rational equations in solving problems

• 2.05 Use rational equations to model and solve
  problems justify results.
• Solve using tables, graphs, and algebraic
  properties.
• Interpret the constants and coefficients in the
  context of the problem.
• Identify the asymptotes and intercepts
  graphically and algebraically.

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• A rational equation is an equation in which one
  or more of the terms is a fractional one. 
• When solving these rational equations, we utilize
  one of two methods that will eliminate the
  denominator of each of the terms.   

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Method 1

• If the equation is in the form of a proportion 
  
• you can use "cross-multiplication" to eliminate
  the denominator, as in  . 
• Then solve the resulting equation and check.

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Examples
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Method 2

• To solve the rational equation in this method,
  we
•   Identify the least common denominator (LCD),
•   Multiply each side of the equation by the LCD,
  
• simplify,
•   Solve the resulting equation, and
• Check the answer.

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Examples
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More Examples
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Application 1

• Carlos can travel 40 mi on his motorbike in
  the same time it takes Paul to travel 15 mi on
  his bike. If Paul rides his bike 20 mi/h slower
  than Carlos rides his motorbike, find the speed
  for each bike.

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Application 2

• A passenger train travels 392 mi in the same
  time that it takes a freight train to travel 322
  mi. If the passenger train travels 20 mi/h
  faster than the freight train, find the speed of
  each train.

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Application 3

• Sidney can paint a fence in 8 hours. Roy can
  do it in 4 hours. How long will it take them to
  do the job if they work together?

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Application 4

• One pump can fill a tank with oil in 4 hours.
  A second pump can fill the same tank in 3 hours.
  If both pumps are used at the same time, how
  long will they take to fill the tank?

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In this chapter you should have

• Learned to use inverse variation and the graphs
  of inverse variations to solve real-world
  problems.
• Learned to identify properties of rational
  functions.
• Learned to simplify rational expressions and to
  solve rational equations.