Section 2.3
Rational Functions and Their Graphs
A rational function can be expressed as f ( x) =
p( x )
where p(x) and q(x) are
q( x )
polynomial functions and q(x) ≠ 0.
Vertical Asymptote of Rational Functions
The line x = a is a vertical asymptote of the graph of a function f if f(x)
increases or decreases without bound as x approaches a.
1
Example: Given the graph of f ( x ) = .
x
Locating Vertical Asymptotes and Holes
Factor the numerator and denominator. Look at each factor in the
denominator.
• If a factor cancels with a factor in the numerator, then there is a hole
where that factor equals zero.
• If a factor does not cancel, then there is a vertical asymptote where that
factor equals zero.
x 2 − 3x − 10
Example 1: Find any vertical asymptotes and/or holes of f ( x) = 2
.
x −x−6
Section 2.3 – Rational Functions
1
Horizontal Asymptote of Rational Functions
The line y = b is a horizontal asymptote of the graph of a function f if f(x)
approaches b as x increases or decreases without bound.
1
Example: Given the graph of f ( x ) = .
x
Horizontal asymptotes really have to do with what happens to the y-values as
x becomes very large or very small. If the y-values approach a particular
number at the far left and far right ends of the graph, then the function has a
horizontal asymptote.
Note: A rational function may have several vertical asymptotes, but only at
most one horizontal asymptote. Also, a graph cannot cross a vertical
asymptote, but may cross a horizontal asymptote.
Locating Horizontal Asymptotes
p( x )
Let f ( x) =
,
q( x )
Shorthand: degree of f = deg(f), numerator = N, denominator = D
1. If deg(N) > deg(D) then there is no horizontal asymptote.
2. If deg(N) < deg(D) then there is a horizontal asymptote and it is
y = 0 (x-axis).
a
,
b
where a is the leading coefficient of the numerator and b is the leading
coefficient of the denominator.
3. If deg(N) = deg(D) then there is a horizontal asymptote and it is y =
Section 2.3 – Rational Functions
2
Example 2: Find the horizontal asymptote, if there is one, of
( 3x )( 2x − 1)
.
f ( x) =
4x 2 − 1
Slant Asymptotes
The graph of a rational function has a slant asymptote if the degree of the
numerator is one more than the degree of the denominator.
To find the equation of the slant asymptote, divide the denominator into the
numerator. The quotient is the equation of the slant asymptote. As x → ∞ ,
the value of the remainder is approximately 0.
The equation of the slant asymptote will always be of the form y = mx + b.
x 3 − 2x 2 − x + 2
Example 3: Find the slant asymptote for f ( x) =
x2 − 4
Section 2.3 – Rational Functions
3
Steps to Graphing a Rational Function
1. Factor numerator and denominator. If a factor in the numerator cancels
with a factor in the denominator then there is a hole in the graph when that
cancelled factor equals zero.
2. Find the x-intercept(s) by setting the numerator equal to zero.
3. Find the y-intercept (if there is one) by substituting x = 0 in the function.
4. Find a horizontal asymptote, if there is one. If not, then see if it has a slant
asymptote
and find its equation.
5. Find the vertical asymptote(s), if any, by setting the denominator equal to
zero.
6. Use the x-intercept(s) and vertical asymptote(s) to divide the x-axis into
intervals. Choose a test point in each interval to determine if the function is
positive or negative there. This will tell you whether the graph approaches the
vertical asymptote in an upward or downward direction.
7. Graph! Except for the breaks at the vertical asymptotes, the graph should be
a nice
smooth curve with no sharp corners.
Section 2.3 – Rational Functions
4
Example 4: Sketch the graph of f ( x) =
Section 2.3 – Rational Functions
2x 2 − 18x
.
x 2 − 11x + 18
5
Example 5: Sketch the graph of f ( x) =
Section 2.3 – Rational Functions
16 − x 4
.
2x 3
6
Note: It is possible for the graph to cross the horizontal
asymptote, maybe even more than once. The method used to
figure out graph of a function crosses (and where), set y equal to
the value of the horizontal asymptote and then solve for x.
Example 6: Does the function cross the horizontal asymptote?
x 2 − 2x − 3
a. f ( x) = 2
x +x−6
b.
f ( x) =
1 − 2x
x+1
Typeequationhere.
Example 7:
Find the correct graph to match the following function.
=
=
Section 2.3 – Rational Functions
=
7
Example 8: Find the correct graph to match the following function.
b
a.
=
Section 2.3 – Rational Functions
=
8