Graphing Rational Functions
Objective: You should be able to graph rational functions without a calculator.
Rational Function: a function of the form f ( x) 
p( x)
, where p(x) and q(x) are polynomials and q(x)  0.
q( x)
In this section we are going to learn to graph rational functions for which p(x) and q(x) are linear.
Hyperbola Properties:
*a horizontal asymptote
*a vertical asymptote
*the graph has 2 symmetrical parts called branches.
**If a rational function is in the form y 
a
k
xh
, they are hyperbolas with asymptotes at x = h and y = k.
The domain cannot include h. Similarly, the range cannot include k.
How to graph a hyperbola of the form y 
a
k:
xh
1. Determine the vertical and horizontal asymptotes.
2. Plot 2 points on each side of the vertical asymptote.
3. Draw the 2 branches of the hyperbola that approach both asymptotes (and pass through the plotted
points.)
Example 1: Graph the function. State the domain, range, and asymptote equations.
a) y 
3
2.
x 1
10
x
8
6
4
2
-10 -8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
y
1
3
b) y 
x4
10
8
x
6
y
4
2
-10 -8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
-10
ax  b
cx  d
** If a rational function is in the form y 
, they are also hyperbolas.
The vertical asymptote occurs at the x-value that makes the denominator zero. (Domain: x  the value that
gives you zero in the denominator)
The horizontal asymptote is the line y 
a
c
How to graph a hyperbola of the form y 
(Range: all real #s except
ax  b
cx  d
a
)
c
:
1. Determine the vertical and horizontal asymptotes.
2. Plot 2 points on each side of the vertical asymptote.
3. Draw the 2 branches of the hyperbola that approach both asymptotes (and pass through the plotted
points.)
Example 2: Graph the function. State the domain, range, and asymptote equations.
a) y 
x2
3x  3
10
x
8
6
4
2
-10 -8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
y
4x 1
b) y 
2x  3
10
8
x
6
4
2
-10 -8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
y