Steps To Graph Rational Functions
1. Make sure the numerator and denominator of the function are arranged in descending order
of power.
2. Find the domain
a. Factor the denominator of the function, completely.
b. Find the real zeros of the denominator by setting the factors equal to zero and solving.
c. Write “Domain = {x|x6=}”
3. Factor the numerator and denominator, completely. Cancel any common factors between
the numerator and denominator. (After you have canceled factors, you are left with the
reduced function. Use the reduced function for all remaining steps.) If there are no common
factors, then skip Step 4.
4. Find any Holes
a. Set any common factor found in Step 3 equal to zero and solve.
This is the x-coordinate for the hole.
b. Substitute this x-value back into the reduced function found in Step 3.
This is the y-coordinate for the hole.
b. Plot the hole on the graph as an open circle.
5. Find any Vertical Asymptotes
a. Set each factor from the denominator of the reduced function equal to zero and solve.
b. Write each equation in the form x= .
c. Draw each vertical asymptote as a dotted line on the graph.
6. Find the Horizontal Asymptote (or Slant Asymptote)
Refer to the reduced function. You will need to know the leading terms from the numerator and the
denominator. That is,
F(x) = anxn+. . .
bmxm+. . . ,
where the degree of the numerator is nand the degree of the denominator is m.
i. If n < m, then the x-axis (given by y= 0) is the horizontal asymptote of the graph.
ii. If n=m, then the line y=an
bm
is the horizontal asymptote of the graph.
iii. If n > m, then the graph has no horizontal asymptote.
The graph will only have a slant asymptote if n=m+ 1. To find the slant asymptote, use long
division. The slant asymptote will be the line given by y=quotient.
quotient
divisor´dividend
