RATIONAL FUNCTIONS

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SECTION 3.2

• RATIONAL FUNCTIONS

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RATIONAL FUNCTIONS

• Rational functions take on the form

Where p(x) and q(x) are polynomials and q(x) ¹ 0
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EXAMPLES
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DOMAIN OF RATIONAL FUNCTIONS

• ANY VALUE WHICH ZEROS OUT THE DENOMINATOR MUST BE
  EXCLUDED FROM THE DOMAIN.
• Do Example 1

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GRAPHS OF RATIONAL FUNCTIONS

• ASYMPTOTE - a line which the graph will approach
  but will never reach.

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HORIZONTAL ASYMPTOTES

• When the degree in the numerator is equal to the
  degree in the denominator.
• Example Graph

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• By studying the graph, we can describe what is
  happening using the following symbols
• As x , f(x) 3
• As x - , f(x) 3
• We say f(x) has a horizontal asymptote of y 3.

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To see the algebraic reasoning behind this,
divide the rational expression through by x2 and
examine what happens as x gets huge.
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The numerator goes to 3 and the denominator goes
to 1.
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EXAMPLE
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Again, algebraically, we can see that the
numerator goes to 4 and the denominator goes to 1
as x gets huge. Thus, this function has a
horizontal asymptote y 4.
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• We can get even more specific with our symbols
  when describing the graph of the function and say
  the following
• As x , f(x) 4 -
• As x - , f(x) 4

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QUESTION
Can you find an easy way of looking at the
symbolic form of a rational function in which the
degree in the numerator is equal to the degree in
the denominator to find the horizontal asymptote?
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(No Transcript)
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• Answer
• Just divide the leading coefficient in the
  numerator by the leading coefficient in the
  denominator.

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EXAMPLE

• Determine the equation of the horizontal
  asymptote for the graph of

Horizontal Asymptote y 3/2
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VERTICAL ASYMPTOTES

• Vertical lines are always given by an equation in
  the form x c. Here, the graph of the rational
  function will approach a vertical line, yet never
  quite reach it. This means that this is a value
  x will never equal.

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• Vertical asymptotes are nothing more than domain
  restrictions, or values for the variable that
  will cause the denominator to equal 0.

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EXAMPLE
x 2 is not in the domain of f(x) because
replacing x with 2 would cause the denominator to
equal 0. Graph f(x).
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• Here, we can describe what is happening to the
  graph of the function by using the following
  language
• As x 2 - , f(x) -
• As x 2 , f(x)

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• Thus, the vertical asymptote for this function is
  x 2.
• Is this the only asymptote for this function?
• No! This function also has a horizontal
  asymptote given by the equation y 0

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• As x gets huge, the numerator goes to zero and
  the denominator goes to 1.

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• In fact, any rational function in which the
  degree in the numerator is less than the degree
  in the denominator will have a horizontal
  asymptote of y 0 (or the x-axis).

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EXAMPLE
We can either examine the graph to determine the
asymptotes, or we can study the symbolic form,
using the tools weve learned.
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• The vertical asymptotes will be the domain
  restrictions. What values will zero out the
  denominator?

Thus, the only vertical asymptote is x 2.
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• Now, for the horizontal asymptotes.
• The degree in the numerator is equal to the
  degree in the denominator.
• Thus, the horizontal asymptote is y 3. Graph
  the function.

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EXAMPLE

• Determine all asymptotes

H.A. y 4
V.A. x - 4 x 2
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SLANT ASYMPTOTES

• When the degree in the numerator is exactly one
  more than the degree in the denominator.

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To determine the slant asymptote, we simply do
the division implied by the fraction line. We
can use synthetic division.
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• 2 1 - 3 6

2
- 2
1
- 1
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Thus, f(x) can be written as
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• As x gets huge, the fractional part tends toward
  0 and the entire function tends toward the linear
  function y x - 1.
• Graph the function.

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EXAMPLE

• Determine all asymptotes of the function below

Vertical Asymptotes x - 3 x 2
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x2 x - 6

• Slant Asymptote

2x
1
2x3 2x2 - 12x
x2 12x
11x 6
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Slant Asymptote y 2x 1 Graph the function
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• CONCLUSION OF SECTION 3.2