Chapter 8 Rational Functions

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Chapter 8 Rational Functions
Definition A rational function f is a quotient
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 function f(x) g(x)/h(x) g(x) h(x)
polynomials continuous(smooth,no
breaks)/discontinuous (if h(x)0)
vertical asymptotes if 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. slant asymptotes degree of denom is
1 or higher numerator is exactly 1 more use
long division to get equation of the line for the
slant removable discontinuity/hole common
linear factor (cancels) number line test
plot all factors (num denom.) test values of x
to see if f(x) is /- 8.4 Simplifying/Multiplyin
g Dividing 8.5 Adding Subtracting (common
denominator) 8.6 Solving Rational Equations
cross multiplying 8.7 Probability of Multiple
Events (and multiply or add)
mutually exclusive (no outcome is repeated in the
events) not mutually excl. (must subtract
repeats so P(A B) P(A) P(B) P(repeats)
indep./depend.(removal of 1st choice changies
options for 2nd )
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Advanced Math Warm up Name___________After C7
Test

• Functions equation shape
• (write using h k)
• 1. Linear
• a. Slope-intercept form
  ______________
• b. Special case constant ______________
• c. Special case direct variation ______________
• Absolute value ______________
• Quadratic ______________
• Square root _______________

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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
  if k _____ then _________
• if k _____ then _________
• Examples
• 1. Given - X 0.5 2 6
• Y 1.5 6 18
• Whats the relationship?
• 2. Determine whether inverse or direct variation.
  Write equation.
• X 0.2 0.6 1.2
• Y 12 4 2

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• Name________________________
• Advanced Math Warm-Up after 8.1
• Identify the data as a direct variation or an
  inverse variation. Then write an equation to
  model the data.
• 1. 2.
  3. 4.
  
• 5. Create a translated absolute value function
  and
• sketch it without a calculator.
• Name________________________

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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
C8 Rational Functions Special rational
function is Inverse Variation _______
Translated Inverse
variation function y a__ k
x-h Which
moves the basic function h units __________
k units __________. When a the functions graph
s (__________) are in the ________
quadrants. When a the branches are in the
_____________ quadrants. The invisible lines the
branches approach are called ________________.
The vertical asymptote is at ________ and the
horizontal asymptote is at ________.

• domain a function must be defined for all
  values of its domain a function is not defined
  for values of x that make the denominator 0.
• Examples Identify the horizontal and vertical
  asymptotes. Graph b d.
• a. Horizontal asym. ____________
  vertical asym ____________ b.

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• Write an equation for a translation of y 4/x
  with the given asymptotes,
• a. X 2 and y 0
• b. X -3 and y -1
• X -2 and y 3
• X 3 and y -4
• Homework Practice 8-2 in workbook 3-14 (just
  identify vertical and horizontal asymptotes
  dont graph ) and 15 - 26

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• Name________________________
• Advanced Math Warm-Up after 8.2
• Identify the data as a direct variation or an
  inverse variation. Then write an equation to
  model the data.
• 1 Create a translated quadratic function and
• sketch it without a calculator.
• Graph
• Write an equation for
  translated down 2 and left 4. ________
• Name________________________
• Advanced Math Warm-Up after 8.2

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8.3 Rational Functions and their Graphs

• Rational Functions Given g(x) and h(x) are
  _________ functions, then
• is a ______________ function.
• Examples
• 1. 2.
  3.
• Note Polynomial functions (like linear
  functions, quadratic and cubic functions) are
  _______________ curves they have no ______,
  _______, or _______. HOWEVER the graph of a
  rational function may be ___________ ! (It
  might have ________, _______, or __________.)
• Continuous or Discontinuous? A rational function
  is discontinuous at the real values of ___ which
  make it ______. ( cant divide by 0) Find
  discontinuities by setting ___ _____.
• Discontinuities are either ________________ or
  _________. If the denominator discontinuity has a
  common factor in the numerator, then it is
  removed when simplified. This ________
  ________ creates a ________. All other
  discontinuities are __________.
• Is the function continuous or discontinuous? If
  discontinuous which type?

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8.3 Rational Functions and their Graphs

• Identify horizontal asymptotes by ___________ the
  polynomials. The quotient is the __________
  ____________.
• Examples ( Find the horizontal asymptotes)
• 6. 7.
  8.
  
  
• Notice
• degree num degree denom.
• degree num lt degree denom.
• degree num gt degree denom.
• when degree of denom. Is 1 or higher num. is
  one higher
• Graphing by hand
• Example 10 Sketch the graph of
• Step 1 Find the vertical asymptotes holes
  (set the bottom equal to zero)

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8.3 Rational Functions Their Graphs

• Examples Graph by hand
• 11. y 4x 2
• x 3
• 12. y (x-2)(x2)
• x 2
• 13. y x² 4
• 3x 6
• Graphing with the Calculator
• Example The CD-ROMs for a computer game can be
  produced for 0.25 each. The development cost is
  124,000. The first discs are samples and will
  not be sold.
• a. Write a function c(x) for the average
  cost of a saleable disc.
• b. Graph the function using the calculator.

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Advanced Math Name___________________
8.3 Homework Warm Up

• Classify each function as continuous or
  discontinuous. If discontinuous, give the values
  of x for which the function is undefined.
• f(x) set the denominator 0
  solve for x
• if x imaginary
  number then the function is continuous
• if x real number then the
  function is discontinuous at that value of x
• f(x)
• 4. f(x)
• Graph the function. Include any horizontal,
  vertical asymptotes, and holes.
• 11. f(x)
• find denominator zeros (set 0 solve) -
  discontinuities
• if the zeros factor can cancel out with
  the numerator hole
• otherwise its a vertical asymptote

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Advanced Math Name___________________
8.3 Homework Warm Up

• Graph the function. Include any horizontal,
  vertical asymptotes, and holes.
• 12. f(x)
• denominator zeros
• degree of num vs denom
• numerator zeros for x intercepts
• y intercept ( plug in x 0)
• sign chart plotting zeros of num and denom
• example f(x)

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Advanced Math Name___________________
8.3 part 2 Homework Warm Up

• Classify each function as continuous or
  discontinuous. If discontinuous, give the
  value(s) of x for which the function is
  undefined.
• 1.
• 2.
• 3.
• 4.

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Advanced Math Name___________________
pledge______________Quiz 8.3

• Graph the given functions.
• f(x) 3x 2. f(x) ?x - 3? 1
• 3. f(x) (x2)² - 3 4. f(x)
• f(x) x³ 2 6. f(x)
• f(x)
  8. f(x)

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• Given the point (7, 4) is from a set of data that
  varies inversely, find the constant of
  variation. _______
• Given the point ( 4, 8) is from a set of data
  that varies directly, write an equation to model
  the variation. _______
• Identify the asymptotes of the following inverse
  variations.
• 13. vertical asymptote
  _____________ horizontal asymptote __________
• 14. vertical asymptote
  _____________ horizontal asymptote __________
• Write an equation for a translation of y 4/x
  with the given asymptotes
• x 3 and y 2 ______________________
• x -4 and y 1 ______________________

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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)
• First factor the expressions. Then simplify
  expressions by canceling out common factors.
  Terms CANNOT be canceled out!
• Factors are connected by ______ and can be
  canceled.
• Terms are connected by _______ and CANNOT be
  canceled.
• Examples Factor and Simplify.

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Advanced Math Name___________________
8.4 Homework Warm Up

• Simplify the rational expression.
• 1.
• 2.
• 3.
• 4.
• 5.

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Advanced Math Name___________________
8.4 Homework Warm Up

• Simplify the rational expression.
• 1.
• 2.
• 3.
• 4.
• 5.

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8.5 Adding Subtracting Rational Functions

• When adding and subtracting functions, you must
  get a ______ denominator. You can do this by
  finding the _________ _________ __________
  (LCM) of the denominators.
• Example 1 Suppose an object is 15 cm from a
  camera lens. When the object, seen through the
  lens, is in focus, the lens is 10 cm from the
  film. Find the focal length of the lens.
• The lens equation is
  where f is the focal length
• represents distance from
  lens to film
• 
  represents distance from
  lens to object
• Example 2 Find the LCM of each pair of
  polynomials.
• a.

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8.5 continued

• Example 3 Simplify.
• a.
• b.
• c.
• Example 4 Simplify.
• a.

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8.6 Solving Rational Equations

• First type _________________ Solve by
  __________________
• a. b.
  c.
  .
• Second type________________ Solve
  by___________________
• a. b.
  c.
  .
• Example 3 Your company makes ecology posters.
  The office expenses are 54,000 a year. The
  materials for each poster cost 0.28. The
  company can produce and sell twice as many
  posters next year as this year. This will reduce
  the per poster cost by 1.8. How many posters
  are you producing this year?
• Step 1 Define the variables.
• Step 2 Relate the variables in an
  equation.