Graphing Rational Functions
To graph a rational function, we’ll need to find the
following items:
• x-intercepts (where the graph will touch or cross
the x-axis)
• y-intercepts (where the graph will touch or cross
the y-axis)
• Vertical, horizontal and/or oblique asymptotes
(what lines bound the graph)
• Where the function is positive or negative
(dependent on the x-intercepts and the vertical
asymptotes)
Term
x-intercept
Explanation
Typically, to find the x-intercept algebraically, we would
1. Set
(or f(x)) and
2. Solve for x.
For a rational expression, we can set the numerator
equal to zero and solve for x to find the x-intercepts.
Proof/Description
Example
If we set
or
, the
following equation will be the result:
Set the numerator equal to zero and
solve for x.
If we multiply both sides of the
equation by the denominator to “clear”
the fraction, we would be left with
Therefore, to find the x-intercepts of a
rational expression, set the numerator
equal to zero and solve for x.
1
For a polynomial,
You can solve for x by factoring the
polynomial
Or using the quadratic equation1.
The two x-intercepts are
and
.
, you can solve for x using the quadratic equation,
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Graphing Rational Functions
Term
y-intercept
Explanation
Proof/Description
Typically, to find the y-intercept algebraically, we would
1. Set
and
2. Solve for y e.g. f(x).
For a rational expression, the y-intercept will be
calculated by the constants in the numerator and the
denominator.
Example
If we set
, the following equation
will be the result:
For the y-intercept, it would be equal
to the constant in the numerator
divided by the constant in the
denominator.
Therefore, we would be left with
Vertical
Asymptotes
The vertical asymptotes indicate where the function is
undefined.
For a rational expression, the function is only undefined
where the denominator is equal to zero.
Remember, you CANNOT divide by zero.
Therefore, to find the vertical asymptotes, set the
denominator equal to zero and solve for x.
Therefore, the y-intercept would be
equal to the constant in the numerator
divided by the constant in the
denominator.
The vertical asymptotes are indicated
by vertical dotted lines on the graph.
The graph of the function will be
bounded by the line. Basically, the final
graph will may come close to but will
not touch the vertical asymptote.
Set the denominator equal to zero and
solve for x to find the vertical
asymptote.
There are 2 vertical asymptotes
and
.
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Graphing Rational Functions
Horizontal &
Oblique
Asymptotes2
The horizontal and oblique asymptote also bounds the
graph.
To consider if you’ll have a horizontal or oblique
asymptote, we want to look at the largest term in the
numerator and the denominator.
The horizontal and oblique asymptotes
are similar to the vertical asymptote in
that they are indicated by a dotted line
and the final graph may come close to
but will not touch either asymptote.
If the degree of the largest term in the numerator (n) is
equal to the degree of the largest term in the
denominator (m) or if the degree of the denominator
(m) is greater than the degree of the numerator (n),
we’ll have a horizontal asymptote.
If the degree of the numerator (n) is one more than the
degree of the denominator (m), we will have an oblique
asymptote.
Horizontal Asymptote, H.A.
or
H.A: .
Example 1:
The degree of the numerator (2) is
equal to the degree of the
denominator (2), only a horizontal
asymptote exists
H.A.
, the x-axis
Example 2:
H.A: .
Oblique Asymptote, O.A.
The line indicated by
is the oblique asymptote.
Find the O.A. by
dividing the
numerator by the
denominator.
The degree of the numerator (2) is less
than the degree of the denominator
(3), therefore the horizontal
asymptote is the x-axis.
H.A.
Example 3:
Since the degree of the numerator (3)
is one greater than the degree of the
denominator (2), there is only an
oblique asymptote.
To calculate the oblique asymptote,
divide the numerator by the
denominator.
2
NOTE: You will either have a horizontal asymptote, an oblique asymptote or neither.
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Graphing Rational Functions
Determining
where the
Graph is
above/below
the x-axis
3
Draw a line representing the x-axis and indicate the
points where an x-intercept exists as well as where
vertical asymptote exists.
Pick a test point (any point) before and after each xintercept or vertical asymptote.
Calculate if the function is positive or negative at that
point.3
• If the point is positive, the graph is above the xaxis.
• If the point is negative, the graph is below the xaxis.
You want to pick test points that will
make it easier for you to calculate if the
function is positive or negative at that
point.
If the numerator and denominator are
factored, performing the calculation
can be straightforward.
If the polynomial is not factored, see if you can factor it quickly. If not, calculate the value of the function at that point e.g. at x=a, calculate f(a).
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