Hints For Graphing

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Chapter 3

• Section 3.1
• Hints For Graphing

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ce

• Graph the function
• f(x) x3
• This is a cubic function, which is an example of
  an odd function.
• This function is symmetric through the origin.

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In general

• The graph of a function y f(x) is said to be
  symmetric through the origin if for all x in the
  domain of f, we have f(-x) -f(x)

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Translation

• Translation of y x3 can be applied as with
  quadratics.
• Ce Graph y (x-3)3
• Ce graph y (x3)3
• Ce graph y (x3)3 2
• Ce graph y (x-3)3 2

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Vertical expansionVertical contraction

• Ce graph y x4
• Ce graph y 2x4
• Ce graph y .5x4

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Page 179

• Exercises 1-15 all
• 18 - 20

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Ch 03 Section 3.2

• Rational functions a function that is a ratio
  of polynomials.
• Example
• f(x)

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Ce

• Graph f(x)

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Asymptotes

• Lines that a curve approaches but never touches.

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Horizontal Asymptotes

• The line y k is a horizontal asymptote for the
  graph of a function if

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Vertical Asymptote

• The line x c is a vertical asymptote for the
  graph of a function if
• As x approaches c from either the left or the
  right.

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To find the vertical asymptotes of a function

• Factor numerator and denominator
• Solve the denominator for x.
• These are you vertical asymptotes if the
  denominator is zero but the numerator is not zero.

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Ce

• Graph
• Find the horizontal asymptotes

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Ce

• Sketch the graph and find the horizontal
  asymptotes
• f(x)

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Ce

• Graph
• Find the horizontal asymptotes

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Ce

• Graph
• Find the horizontal asymptotes, domain, and range

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Basic Curves

• You should be able to sketch each of the
  following functions

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Page 188

• 3,7,11,13,15,23

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Sections 3.3

• Polynomial and Rational Functions

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Steps for Graphing Polynomial

• 1. Factor the polynomial if possible
• 2. Find y intercepts
• 3. Find the x intercepts

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• 4. Form a table of signs for f(x)
• Choosing convenient test values for each
  interval. Page 191
• 5. Sketch the graph using the x intercepts, the
  signs of x, and the y intercepts.

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CeUse table of signs to determine the signs of f
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CeUse table of signs to determine the signs of f
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Ce sketch the function
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Ce Sketch the function
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Ce Sketch the function
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Page 197 1 - 7
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Rational Functions and AsymptotesTo Find the
asymptotes follow the following rules.
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Vertical Asymptotes

• If c is a value for which the function is not
  defined, then x c is a vertical asymptote.
• Factor denominator set equal to zero, solve for x

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• Horizontal Asymptote
• If n lt m, then y 0 is a HA
• If n m, then y is a HA
• If n gt m, there are no HA

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Ce

• Find the asymptotes for each rational function
• A.
• B.

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Ce Find the asymptotes for the following
function.
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Ce Find the asymptotes for the following
function.
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Test Your Understanding

• Page 193
• 1-6

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Graphing Rational Functions

• Step 1. Factor the numerator and denominator if
  possible.
• Step 2. Find the x and y intercepts if any
• Step 3. Find the vertical asymptotes.
• Step 4. Find the horizontal asymptotes.
• Step 5. Use a table of signs
• Use a table for other values

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CE

• Graph

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CE
38
CE

• Graph

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CE

• Graph

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Page 196

• Problems 9-13 odd, 15 25 odd

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Section 3.4Rational Equations

• With Fractions

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Fraction Review

• Add
• Must find the LCD

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Note

• To find the x intercepts of a graph, you let y
  0 and solve for x.
• To find the y intercepts of a graph, you let x
  0 and solve for y.

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CeFind the x-intercepts
45
Ce

• Solve for x

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ce

• Solve for x

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Proportion Property

• If
• Then ad bc

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Solve for x
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Page 201

• 1- 10

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Synthetic Division

• Synthetic Division is a special case of long
  division.

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Why do we use synthetic division?

• Synthetic Division is used to factor higher
  degree polynomials.

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CEReview Long Division of polynomials
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CEOne more review
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Synthetic division is just a shortcut for long
division

• Arrange variable is reducing order.
• Drop the variables.
• Just use the coefficient
• Note if there is a missing term be sure to
  replace it with a zero.

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CEUse Synthetic division
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CEUse Synthetic Division
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CEUse Synthetic Division
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Practice

• Page 219 1-4

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Page 222

• 1-9 odd
• Check your answers

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In general

• Whenever a polynomial p(x) is divided by x-c we
  have the general formula.
• p(x)q(x)(x-c)r
• Where q(x) is a quotient and r is a constant
  remainder

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Following from the general formula we have the
REMAINDER THEOREM

• If p(x) is divided by x-c, the remainder is p( c
  )

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CE Find the remainder if

• P(x)
• Is divided by x-2

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CEuse the remainder theorem to find the
remainder when

• Is divided by x2

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Practice

• Page 220 1-4

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Page 223

• 25 29 odd

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Factor Theorem

• A polynomial p(x) has a factor x-c iff p(c) 0.

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• That is x-c is a factor of a function p(x) if you
  divide p(x) by x-c and there is no remainder.

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CEShow

• X-2 is a factor of

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CEUse the factor theorem to show

• X1 is a factor of

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CEShow

• A) X3 is a factor of
• P(x)
• B) Factor P(x) completely
• C) Find the Zeros of p(x)

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Page 223

• 31-45 odd
• Check your answers