5.2 Rational Functions and Asymptotes

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5.2 Rational Functions and Asymptotes

• In this section, we will study the following
  topics
• Graphs of rational functions
• Finding domains of rational functions
• Finding horizontal, vertical, and oblique
  asymptotes of rational functions

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What is a Rational Function?
Like a rational number, which is a number that
can be expressed as a ratio of two integers (a
fraction), a RATIONAL FUNCTION IS A RATIO OF TWO
POLYNOMIALS IN TERMS OF A SINGLE VARIABLE.
E.g. Note that the value of the
denominator cannot equal zero since division by
zero is undefined.
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The Domain of a Rational Function
The domain of a rational function of x includes
all real numbers EXCEPT those values of x that
make the denominator zero. TO FIND THE DOMAIN
OF A RATIONAL FUNCTION, set the denominator equal
to zero and solve for x to determine which
x-value(s) must be EXCLUDED from the domain.
ExampleFind the domain of each function.
Solution
f(x) is undefined when x2 7x 6 ___ g(x) is undefined when x ___
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The Graph of a Rational Function Vertical
Asymptotes
Example
The numerator must be placed inside parentheses since the entire expression is being divided by x.
Notice that the graph has two branches. Also, notice how the graph seems to get closer and closer to the vertical line x 0 but never touches it. Remember that f(x) is not defined at x 0.
The vertical line that the graph approaches is
called a _____________________
____________________.
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The Graph of a Rational Function Horizontal
Asymptotes
Example (continued)
Also notice that the graph seems to approach an invisible horizontal line as x heads towards positive and negative infinity. This line is called the _____________________ ___________________. Horizontal asymptotes of rational functions can be found algebraically, but for now we will use our graphing calculators to find it numerically.

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The Graph of a Rational Function Horizontal
Asymptotes
Example (continued)
One way to do this is to use your table. Look at the table on the left. I changed TABLE SETUP to Indpnt Ask instead of Auto. Now I can enter whatever values of x I would like into the function. I want to see what happens as x gets very large and as x gets very small.

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The Graph of a Rational Function Vertical and
Horizontal Asymptotes
Example (continued)
First I will look at increasingly larger values of x (heading towards ?) to see what the functional value (the y-value) is getting closer to. As I enter larger and larger values of x, the y-value seems to get closer and closer to 2 (from above). Now I will look at what happens as x gets increasingly smaller (heading towards - ?). As I entered smaller and smaller values of x, the y-value again seems to get closer and closer to 2 (from below).
The horizontal asymptote in this example is the
line _____________.
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The Graph of a Rational Function Range
Example (continued)
Look back at the graph of f(x). If you picture a
dotted horizontal line passing through at y 2,
you can see how the branches of the graph get
really, really close to this line as x gets very
large () and as x gets very small (-), one from
above and one from below, but they never reach
the line.
We can also see by this graph that as x gets
closer to zero from the right, the y-values get
larger and head towards ?. As x gets closer to
zero from the left, the y-values get smaller and
head towards - ? . From this, we can determine
the range for f(x). The RANGE would be the set of
all real numbers except 2, which in interval
notation would be ____________________________.
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The Graph of a Rational Function Vertical and
Horizontal Asymptotes
Example Use the graph of the functions to locate
any vertical and horizontal asymptotes. a)
b)

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Finding Vertical Asymptotes Algebraically
The vertical asymptotes of a rational function in
simplest form exist at the x-values for which the
function is undefined. (Remember, the function
is undefined at the x-values for which the
denominator is equal to zero.) So, to find the
vertical asymptotes algebraically, set the
denominator equal to zero and solve for x. (We
used this technique when we found the domain of
the function.)
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Finding Vertical Asymptotes Algebraically
Example
Solution
The function is undefined when x _____ or x
_____ Therefore, the domain of f(x) is
________________________
The vertical asymptotes are the lines ________
and _________
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Finding Horizontal Asymptotes Algebraically
To determine the horizontal asymptote, we compare
the degrees of the numerator and denominator of
the rational function.
Horizontal Asymptotes of a Rational Function Let f(x) be the rational function where n is the degree of N(x) and d is the degree of D(x) If n lt d, the horizontal asymptote of the graph of f is the line y 0. If n d, the horizontal asymptote of the graph of f is the line is If n gt d, the graph of f has NO HORIZONTAL ASYMPTOTE. (The graph may have an oblique asymptotemore on this soon.)
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Finding Horizontal Asymptotes Algebraically
Example Identify any horizontal asymptotes of the
graphs of the functions.
a) b) c)

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Choose the graph of the equation
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Oblique (Slant) Asymptotes
We just learned that if the degree of the
numerator is greater than the degree of the
denominator, the graph has no horizontal
asymptote. Taking this one step further, if the
degree of the numerator is exactly ONE greater
than the degree of the denominator, the graph has
an OBLIQUE (SLANT) ASYMPTOTE. Example

The graph of f has an oblique asymptote since
the degree of the numerator is 2 and the degree
of the denominator is 1.
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Oblique Asymptotes

• To find the equation of an oblique asymptote, you
  use LONG DIVISION to divide the numerator by the
  denominator.
• The quotient will give you the equation for the
  slant asymptote (Ignore the remainder.)
• The equation of the oblique asymptote will have
  the form y mx b.
• Note A graph of a rational function that has an
  oblique asymptote will not have a horizontal
  asymptote but may have a vertical asymptote.
• You sketch the graph of a rational function with
  an oblique asymptote following the same
  guidelines I listed before, with the addition of
  using a dashed line to draw the oblique
  asymptote.

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Oblique Asymptotes
Example In the previous example, the oblique
asymptote is found by dividing numerator by
denominator

The quotient gives the equation of the oblique
asymptote, namely, y 2x 11.
This is how you write your answer.
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• End of Section 5.2
• Quote of the Day Rational functions are a pain
  in the asymptote!