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Graphing Rational Functions
What is a rational function? 11 2 2 0
1 2 2 1 0
11 2 2 01 2 2 1 0
...( )
...
n n nn n n
m m mm m m
a x a x a x a x a x a xf x
b x b x b x b x b x b x
or
any polynomial( )
another polynomialf x
Creating the graph
1. Put a vertical asymptote through the non-removable zeros of the denominator. (note: removable discontinuities get covered in calculus)
2. Plot any horizontal or slant asymptotes.3. Plot as many points as needed between and
beyond asymptotes to determine the shape of the graph.
4. Sketch the curve.
Plotting points beyond and between
asymptotes • It will work to pick any number you want between
and beyond asymptotes. A superior student will include the x and y intercepts
• x intercepts occur where y=0. For rational expressions this is where the numerator equals zero.
• y intercepts occur where x = 0. Just plug in a zero.
Plot any vertical asymptotes
• Remember that division by zero is not allowed.• Put a vertical asymptote through any number that
makes the denominator (bottom) zero. You can think of it as a fence that says “Don’t go here.”
• Advanced: If a factor in the denominator can be canceled, it’s called a “removable discontinuity.” You mark it with a hole in the graph rather than an asymptote.
Plot any horizontal or slant asymptotes
• Compare the degrees of the numerator (n) and the denominator (m).
• If n < m, you have the x-axis for a horizontal asymptote.
• If n = m, you have a horizontal asymptote that is a fraction of the leading coefficients.
• If n = m + 1, then you use long division to find the slant asymptote.
n
m
ayb
Example 1, step 13
1y
x
Put a vertical asymptote through the zeros of the denominator.
In this example, the denominator is zero when x = -1
Example 1, step 23
1y
x
Compare the degrees of the numerator and denominator. Use the chart to decide which type of asymptote (if any) you have.
n = 0
m = 1
Since the degree in the numerator is smaller, the graph has a horizontal asymptote on the x-axis
Example 1, step 33
1y
x
Plot as many additional points as needed to determine the shape of the graph.
We need one point to either side of x = -1 to decide which way the graph will go.
Let’s use x = 0 and -2
33
0 1(0,3)
33
2 1( 2, 3)
Example 1, step 43
1y
x
Now we sketch the curve.
-10 10
-10
10
x
y
-10 10
-10
10
x
y
-10 10
-10
10
x
y
Example 24 12
2 3
xy
x
3. Find the intercepts. (0, -4) and (-3, 0)
1. Find the vertical asymptote.
3
2x
2. Find the slant/horizontal asymptote
42
2y
4. Find a point on the right side
x =2 gives y = 20 which is off the graph but tells us which way it goes5. Sketch the
curve
-10 10
-10
10
x
y
-10 10
-10
10
x
y
-10 10
-10
10
x
y
Example 32 20
2
x xy
x
1. Graph the vertical asymptote
x = 2
Example 3 continued2 20
2
x xy
x
2. Since the degree in the numerator is 1 higher than the degree in the bottom (n = m + 1), we use long division to find a slant asymptote.
2
2
32 20
2
3 20
3 6
14
xx x x
x x
x
x
The top will always be a y = mx + b equation. Discard the remainder.
-10 10
-10
10
x
y
Example 3 continued
-10 10
-10
10
x
y
-10 10
-10
10
x
y
3. Find the x and y intercepts.
You will need to factor the numerator to find the x-intercepts.
5 4
2
x xy
x
(0, 10), (-5, 0), (4, 0)
4. Draw
-10 10
-10
10
x
y
-10 10
-10
10
x
y
-10 10
-10
10
x
y
-10 10
-10
10
x
y
Example 4
2
5
4y
x
3. Find the intercepts.
1. Find the vertical asymptotes.
2. Find the horizontal/slant asymptote
4. Plot extra points to determine the shape of the graph. You will need one on each side of the asymptotes as well as between the asymptotes and intercepts. Choose x = 1, -1, 3, -3
5. Sketch.
-10 10
-10
10
x
y
-10 10
-10
10
x
y
-10 10
-10
10
x
y
Example 5 (last one)
2 9
xy
x
3. Plot your intercepts.
1. Find the vertical asymptotes.
2. Find the horizontal/slant asymptotes.
4. Plot as many extra points as needed between and beyond asymptotes. Choose x = 1, -1, 4, and -4
5. Sketch.
-10 10
-10
10
x
y
