Objectives
Rational Functions:
Asymptotes
REVIEW
Find the domain of f ( x) =
Rational Function
A rational function is a function of the form
f ( x) =
– Find domain of rational functions.
– Use transformations to graph rational functions.
Use arrow notation.
– Identify vertical asymptotes.
– Identify horizontal asymptotes.
– Identify slant (oblique) asymptotes.
P ( x)
Q( x)
Where p and q are polynomials and p(x) and
q(x) have no factor in common and q(x) is not
equal to zero.
Solution:
When the denominator
x + 4 = 0, we have x = −4,
so the only input that
results in a denominator of
0 is −4. Thus the domain
is (−∞, −4) ∪ (−4, ∞).
REVIEW: The graph of
the function is the graph
of y = 1/x translated to the
left 4 units.
Example
Arrow Notation
Find the domain of each rational function.
x 2 − 25
a) f ( x) =
x −5
x
b) g ( x) = 2
x − 25
1
x+4
x+5
c) h( x) = 2
x + 25
Symbol
Meaning
x → a+
x approaches a from the right
x → a−
x approaches a from the left
x→∞
x approaches infinity; x increases without bound
x → −∞
x approaches negative infinity; x decreases
without bound
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Vertical Asymptotes
Rational Functions
Different from other functions because they
have asymptotes.
Asymptotes- a line that the graph of a
function gets closer and closer to as one
travels along that line in either direction.
Vertical Asymptote
Occurs where the function is undefined,
denominator is equal to zero.
Form: x = a
where a is the zero of the denominator
**Graph never crosses **
Vertical asymptotes
• Look for domain restrictions. If there are values of
x which result in a zero denominator, these values
would create EITHER a hole in the graph or a
vertical asymptote. Which?
– If the factor that creates a zero denominator cancels
with a factor in the numerator, there is a hole. If you
cannot cancel the factor from the denominator, a
vertical asymptote exists.
• If you evaluate f(x) at values that get very, very
close to the x-value that creates a zero
denominator, you notice f(x) gets very, very, very
large! (approaching pos. or neg. infinity as you get
closer and closer to x)
Example
Find the vertical asymptotes, if any, of each rational function.
a) f ( x) =
x
x −1
2
b) g ( x) =
x −1
x2 − 1
c) h( x) =
x −1
x2 + 1
Try
2x − 3
this. f ( x) = x 2 − 4
• Determine the vertical
asymptotes of the function.
• Factor to find the zeros of
the denominator:
• Thus the vertical
asymptotes are the lines:
Horizontal Asymptotes
Horizontal Asymptote
Determined by the degrees of the
numerator and denominator.
Form: y = a
(next slide has rules)
** Graph can cross **
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Example
Determining if Horizontal Asymptote Exists
Find the horizontal asymptote, if any, of each rational function.
Look at the rational function r ( x) =
P ( x)
Q ( x)
a) f ( x) =
9 x2
3x 2 + 1
b) g ( x) =
9x
3x 2 + 1
c) h( x) =
9 x3
3x 2 + 1
• If degree of P(x) < degree of Q(x),
horizontal asymptote y = 0.
• If degree of P(x) = degree of Q(x), leading coeff P(x) a
= n
horizontal asymptote y =
leading coeff Q(x) bn
• If degree of P(x) > degree of Q(x),
no horizontal asymptote
Example
Find the horizontal asymptote:
6 x 4 − 3x 2 + 1
f ( x) = 4
9 x + 3x − 2
Oblique Asymptote
Degree of p(x) > degree of q(x)
To find oblique asymptote:
Remember:
• The graph of a rational function never
crosses a vertical asymptote.
• The graph of a rational function might cross
a horizontal asymptote but does not
necessarily do so.
What is the equation of the oblique asymptote?
4 x 2 − 3x + 2
f ( x) =
2x +1
– Divide numerator by denominator
– Disregard remainder
– Set quotient equal to y (this gives the
equation of the asymptote)
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Example
Find the slant (oblique) asymptote for
2 x2 − 5x + 7
f ( x) =
x−2
ASYMPTOTE SUMMARY
Occurrence of Lines as Asymptotes
• For a rational function f(x) = p(x)/q(x), where p(x) and q(x) have
no common factors other than constants:
• Vertical asymptotes occur at any x-values that make
the denominator 0.
• The x-axis is the horizontal asymptote when the
degree of the numerator is less than the degree of the
denominator.
• A horizontal asymptote other than the x-axis occurs
when the numerator and the denominator have the
same degree.
ASYMPTOTE SUMMARY (cont.)
• An oblique asymptote occurs when the degree of
the numerator is 1 greater than the degree of the
denominator.
• There can be only one horizontal asymptote or
one oblique asymptote and never both.
• An asymptote is not part of the graph of the
function. (You should always include your
asymptotes as dashed lines on graphs that you
draw!!)
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