Chapter 8 Rational Functions

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Chapter 8 Rational Functions
Definition. A rational function f is a
quotient f(x) g(x) / h(x) where g
and h are polynomials
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Chapter 8 Rational Functions
8.1 Inverse Variation y k/x
Direct Variation y kx Constant of
Variation k (test yx y/x) 8.2 Graphing
Inverse Variation branches 2 curves, if
k then in I III quadrant if k
- then in II IV quadrant asymptotes
vertical x horizontal y
translations y k c translation of
yk/x x-b where vertical
asymptote at x b and horizontal at y
c 8.3 Rational Functions their Graph
rational functionf(x) g(x)/h(x) g(x) h(x)
polynomials continuous(smooth,no
breaks)/discontinuous (if h(x)0)
vertical asymptotesif no common factors, when
h(x)0 horizontal asymptotes if same
degree, y a/b if degree numeratorltdegree,
y0 if degree numeratorgtdenomin., no horiz.
asy. removeable discontinuity/hole common
linear factor 8.4 Simplifying/Multiplying
Dividing 8.5 Adding Subtracting (common
denominator) 8.6 Solving Rational Equations
cross multiplying 8.7 Probability of Multiple
Events mutually exclusive
independent/dependent
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Chapter 8 Rational Functions

• Key Terms
• asymptote
• branch
• constant of variation
• continuous
• dependent
• discontinuous
• hole
• independent
• inverse variation
• mutually exclusive
• rational function
• removable discontinuity
• simplest form

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Chapter 8 Rational Functions

• Remember from Chapter 7
• Key Terms
• change of base formula - logb A (log A)/(log
  b)
• common logarithm log with base 10
• compound interest formula AP(1r/n)(nt)
• continuously compounded interest APe(rt)
• decay factor 0ltbgt1
• exponential decay yabx when 0ltblt1
• exponential equation yabx
• exponential function f(x) abx
• growth factor bgt1
• half-life b0.5
• logarithm log
• natural logarithm function ln
• radical equation equation with variable is raised
  to exp.

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Name___________

• Remember What does it look like?
• Functions equation shape
• Linear
• Direct variation 1.a. B.
• Constant 2.a. B.
• Absolute value 3.a. B.
• Quadratic 4.a. B.
• Square root 5.a. B.
• Cubic 6.a. B.
• Exponential
• Growth 7.a. B.
• Decay 8.a. B.
• Rational Functions
• Inverse functions 9.a. B.

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8-1 Exploring Inverse Variation

• Youll learn to
• 1. Identify Inverse Variation given a table of
  data
• 2. Identify Direct Variation given a table of
  data
• 3. Determine the constant of variation given
  the type of variation a point on the curve
• 4. Write an equation given the type of
  variation a point on the curve

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8-1 Exploring Inverse Variation

• Inverse Variation y k/x
• as one variable the other
• Direct Variation y kx
• as one variable the other
• Constant of Variation k
• test data for k yx or y/x

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8-1 Exploring Inverse Variation (continued)

• Examples
• X 0.5 2 6
• Y 1.5 6 18

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8-1 Exploring Inverse Variation (continued)

• Examples
• X 0.5 2 6
• Y 1.5 6 18
• Test kyx .75 12 108
• ky/x 3 3 3
• So y/x 3
• y 3x thus, direct variation

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8-1 Exploring Inverse Variation (continued)

• Examples
• Determine whether inverse or direct variation.
  Write equation.
• X 0.2 0.6 1.2
• Y 12 4 2
• Test k xy
• k y/x

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8-1 Exploring Inverse Variation (continued)

• Examples
• Each point is from a model for inverse variation.
  Find each constant of variation.
• a. (3,7) b. (2.5, 1.5)
• Write an equation for each of the above.
• Each point is from a model for direct variation.
  Find an equation for each.
• a. (4,8) b. ( 7,21)

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8-1 Exploring Inverse Variation (continued)

• Examples
• Heart rates life spans of most mammals are
  inversely related. Write an equation to model
  this inverse variation. Use your equation to
  find the average life span of a lion (heart rate
  76 beats/min.).

Animal heart rate(beats/min) life
span(min) Mouse 634 1,576,800 Rabbit 158 6,
307,200 Horse 63 15,768,000 Cat 126 8,000,0
00
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8-1 Exploring Inverse Variation (continued)

• Examples
• Heart rates life spans of most mammals are
  inversely related. Write an equation to model
  this inverse variation. Use your equation to
  find the average life span of a lion (heart rate
  76 beats/min.).

Animal heart rate(beats/min) life
span(min) Mouse 634 1,576,800 Rabbit 158 6,
307,200 Horse 63 15,768,000 Cat 126 8,000,0
00 Lion 76 ? Squirrel 190 ? Elephant 70yr
s
10yrs 27bpm
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8.2 Graphing Inverse Variation

• Youll learn to
• 1. Identify asymptotes of an inverse function.
• 2. Sketch an inverse function given its
  equation.
• 3. Given a function its translated
  asymptotes, write an equation for the translated
  function.

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8.2 Graphing Inverse Variation

• branch
• branch
• branches 2 curves, if k then in I III
  quadrant
• if k - then in II IV
  quadrant
• asymptotes the lines the graph approaches
• vertical x
• horizontal y

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8.2 Graphing Inverse Variation

• Consider y 4/x, sketch the graph.
• Using your calculator, graph the following,
• vertical asymptote horizontal asymptote
• y 4/x
• y 4/x 2
• y 4/x - 4
• How are the graphs related?

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8.2 Graphing Inverse Variation

• Using your calculator, graph the following,
• vertical asymptote horizontal asymptote
• y 4/x
• y 4
• x - 2
• y 4
• x 4
• How are the graphs related?

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8.2 Graphing Inverse Variation

• Translations of Inverse Variations
• y k c translation of yk/x
• x-b
• where vertical asymptote at x b
• and horizontal asymptote at y c
• Example y 4 - 3
• x 2
• vertical asymptote at x
• horizontal asymptote at y

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8.2 Graphing Inverse Variation

• Translations of Inverse Variations
• y k c translation of yk/x
• x-b
• More Examples
• Sketch y 1/x - 3
• Sketch y - 5 - 4
• x - 2
• Write an equation for a translation of y 4/x
  with the given asymptotes,

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8.3 Rational Functions Their Graphs

• Youll learn to
• 1. Classify a rational function as continuous
  or discontinuous.
• 2. Discern between different discontinuities to
  identify vertical asymptotes removeable
  discontinuities or holes.
• 3. Identify horizontal asymptotes.
• 4. Identify behavior near asymptotes.

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8.3 Rational Functions Their Graphs

• Rational Function
• A function that is the quotient of 2
  polynomials
• f(x) g(x)
• h(x)
• where g(x) h(x) are polynomials
• Examples
• y(x2)(x-1) y -2x y
  1
• (x1)
  (x21) (x2-4)

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8.3 Rational Functions Their Graphs

• Examples
• Y - x 1 2. Y 1
• x2 1
  x22x1
• x2 1 0
• x2 -1
• no real value of x can
• be squared -1, so
• this is continuous

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8.3 Rational Functions Their Graphs

• Examples
• Y - x 1 2. Y 1
• x2 1
  x22x1
• x2 1 0 x2 2x 1 0
• x2 -1 (x 1 )(x 1 ) 0
• x 1 0
• no real value of x can x -1
• be squared -1, so so this one is
• this is continuous discontinuous
• 
  _at_ x -1

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8.3 Rational Functions Their Graphs

• Examples
• Determine if continuous or discontinuous
• Y - x 1 2. Y 1
• x2 1
  x22x1
• Y 1 4. Y x2 1
• x2 16 x2 3
• Y x 1
• (x2 2x 6)

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8.3 Rational Functions Their Graphs

• Examples
• Determine if continuous or discontinuous
• Y - x 1 2. Y 1
• x2 1
  x22x1
• continuous discon. _at_ x -1
• Y 1 4. Y x2 1
• x2 16 x2 3
• discon._at_ 4, -4 continuous
• Y x 1
• (x2 2x 6)
• discontinuous _at_ x -1 /- 7

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8.3 Rational Functions Their Graphs

• Vertical asymptotes
• There may be multiple vertical asymptotes
• if no common factors, when h(x)0
• Horizontal asymptotes
• There is at most one horizontal asymptote
• if same degree, y a/b
• if degree numeratorltdegree denominator, y0
• if degree numeratorgtdegree of denominator, no
  horizontal asymptotes

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8.3 Rational Functions Their Graphs

• Examples
• y 3x 5 y 4x 2
• x 2 x 3
• y (x-2)(x2) y x2 4
• x 2
  3x - 6

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8.3 Rational Functions Their Graphs

• Examples
• y 3x 5 y 4x 2
• x 2 x 3
• discon. _at_ x 2, asym.
• vertical asym. _at_ y 3/1
• y (x-2)(x2) y x2 4
• x 2
  3x - 6

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8.3 Rational Functions Their Graphs

• Examples
• y 3x 5 y 4x 2
• x 2 x 3
• discon. _at_ x 2, asym. discon. _at_ x
  -3, asym.
• vertical asym. _at_ y 3/1 vert. Asym. _at_ y
  4/1
• y (x-2)(x2) y x2 4
• x 2
  3x - 6

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8.3 Rational Functions Their Graphs

• Examples
• y 3x 5 y 4x 2
• x 2 x 3
• discon. _at_ x 2, asym. discon. _at_ x
  -3, asym.
• vertical asym. _at_ y 3/1 vert. Asym. _at_ y
  4/1
• y (x-2)(x2) y x2 4
• x 2
  3x - 6
• discon. _at_ x 2, a hole
• no vertical asymptote
• Degree num. (2)gt denom.(1),
• So no horizontal asymp.
• Graph looks like y x 2 (line)
• with a hole at x 2 (see in table)

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8.3 Rational Functions Their Graphs

• Examples
• y 3x 5 y 4x 2
• x 2 x 3
• discon. _at_ x 2, asym. discon. _at_ x
  -3, asym.
• vertical asym. _at_ y 3/1 vert. Asym. _at_ y
  4/1
• y (x-2)(x2) y 2
• x 2
  3x - 6
• discon. _at_ x 2, a hole disc. At x
  6/32, asym.
• no vertical asymptote degree
  numltdenom.,
• Degree num. (2)gt denom.(1), so horiz.asy. _at_ y
  0
• So no horizontal asymp.
• Graph looks like y x 2 (line)
• with a hole at x 2 (see in table)

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8.3 Rational Functions Their Graphs

• Examples
• Y 1 2. Y 2x 3
• (x 2) x 5
• Y x2 6x9
• x 3

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8.4 Rational Expressions

• Youll learn to
• Simplify rational expressions
• Multiply and divide rational expressions

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8.4 Rational Expressions

• Simplest Form when an expressions numerator
  and denominator are polynomials that have no
  common divisors
• In simplest form Not in simplest form
• X 2 x2
  1/x 2(x-3)
• X-1 x2 3 x x1
  3(x-3)
• You can simplify some expressions by dividing out
  common factors after you factor the expressions.

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8.4 Rational Expressions

• Factor and Simplify,
• X2 10x 25 -27 x3y
• X29x 20 9x4y
• 6 3x 2x2 3x 2
• X2 5x 6 x2 5x 6

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8.5 Adding Subtracting Rational Functions

• 8.5 Adding Subtracting (common denominator)

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8.6 Solving Rational Equations

• 8.6 Solving Rational Equations cross
  multiplying

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8.7 Probability of Multiple Events

• What youll learn
• - Identifying independent and mutually exclusive
  events
• - finding probabilities of multiple events

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8.7 Probability of Multiple Events

• 8.7 Probability of Multiple Events
• mutually exclusive
• independent/dependent
• When the outcome of one event affects the outcome
  of a second event, the two events are dependent.
• When the outcome of one event does not affect the
  outcome of a second event , the two are
  independent.
• When two events cannot happen at the same time,
  the events are mutually exclusive.
• Two situations 1. P(A B) dep/indep
• 2. P (A or B)-exclusive?

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8.7 Probability of Multiple Events

• 8.7 Probability of Multiple Events
• mutually exclusive
• independent/dependent
• Classify as dependent/independent
• Roll a number cube. Then toss a coin.
• Pick a flower from a garden. Then pick another
  flower from the same garden.
• Select a marble from a bag. Replace it select
  another
• Classify dependent as mutually exclusive or not
• Rolling a 2 or a 3 on a number cube
• Rolling an even number or a multiple of 3 on a
  number cube

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8.7 Probability of Multiple Events

• 8.7 Probability of Multiple Events AND
• Independent spinning one pointer, then a
  separate one
• If independent then P(AB) P(A) P(B).
• P(A 1)
• P(B 2)

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8.7 Probability of Multiple Events

• 8.7 Probability of Multiple Events OR
• If mutually exclusive,
• then P(A or B) P(A) P (B)
• If NOT mutually exclusive,
• then P(A or B) P(A) P (B) P(AB)
• A die is rolled. Determine whether each event is
  mutually exclusive or inclusive. Then find the
  probability.
• P(odd or greater than 2)
• P(even or odd)
• P( 6 or 8)