MATH 1031 – Graphing Rational Functions
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To find zeroes (x-intercepts), use numerator of your function. Factor and set = 0. Real zeroes = zeroes
within the real numbers. Imaginary zeroes occur in pairs when you use the quadratic formula and get
a negative number under the square root.
To find vertical asymptotes, use denominator of your function. Factor and set = 0. Your answer will be
in the form of a vertical line, x = a, where a is what makes the denominator zero.
To find the behavior of the graph near each zero, FACTOR! You should have lots of parentheses!
Then, take one zero at a time. Plug it into every x (both numerator and denominator) which will NOT
make it 0. Then, simplify to the get the behavior of the function near that zero. Yes! You will still have
at least one x in your behavior! Be sure to repeat this process for each real zero.
To find horizontal or oblique asymptotes, DO NOT FACTOR! You want NO parentheses! Organize
terms from highest degree to lowest in both the numerator and denominator. We now only care
about the largest degrees in both the numerator and denominator. Let’s call the highest degree of x in
the numerator “n” and the highest degree of x in the denominator “m.”
o If n < m, a horizontal asymptote of y = 0 exists.
o If n = m, a horizontal asymptote exists. To find it, divide the coefficient of the highest degree of
x in the numerator by the coefficient of the highest degree of x in the denominator.
o If n is one greater than m, (n = m + 1), an oblique asymptote exists. Do long division to find it.
(Do not worry about the remainder!)
o If n is two or more greater than m, there is no horizontal or oblique asymptote.
Now, we need to make the chart! Find the important intervals along the x-axis. We know - ∞ and ∞.
What other x-values do we know? Zeroes and vertical asymptotes! If you make a number line to
represent your x-axis, place these x-values in the appropriate location. Choose a number to test in
between each of the significant values. Plug it in as x into your original equation. Your answer is y.
(x,y) is a coordinate on your graph. If y is positive, the graph is above the x-axis on that interval. If y is
negative, the graph is below the x-axis on that interval. Graph your points.
For the graph, pay special attention to if the graph is below/above the x-axis on all intervals. Also, pay
attention to all asymptotes! Remember a graph can NOT cross a vertical asymptote, but it CAN cross a
horizontal or oblique asymptote. To determine if the function does indeed cross a horizontal or
oblique asymptote, set the asymptote equation equal to your original function and solve. If you get x =
something, that is where the graph will cross the asymptote (at x = something). If you get something =
something else and all x’s cancel out, it will not cross.
Finally, on the graph, remember the asymptotes will try to get as close as they can to vertical
asymptotes and other asymptotes in which they won’t cross. This causes the many u and v shapes in
rational graphs.