Intermediate Algebra – Student Workbook – Second Edition 2013

Lesson 6a - Exponents and Rational Functions
Mini-Lesson
RATIONAL FUNCTIONS – KEY CHARACTERISTICS
A RATIONAL FUNCTION is a function of the form
f (x) = p(x)
q(x)
where p(x) and q(x) are polynomials and q(x) does not equal zero (remember that division by
zero is undefined). Rational functions have similar shapes depending on the degree of the
polynomials p(x) and q(x). However, the shapes are different enough that only the general
characteristics are listed here and not a general graph:
VERTICAL ASYMPTOTES
f(x) will have VERTICAL ASYMPTOTES (can be more than one) at all input values where
q(x) = 0. These asymptotes are vertical guiding lines for the graph of f(x) and f(x) will never
cross over these lines.
To find the Vertical Asymptotes for f(x), set q(x) = 0 and solve for x. For each x value that
you find,
x = that value is the equation of your vertical asymptote. Draw a dotted, vertical line on
your graph that represents this equation.
DOMAIN
The x’s you found when solving q(x) = 0 are the values that are NOT part of the domain of
f(x).
To write your domain, suppose that your function denominator is x – a. When you solve x –
a = 0, you get x = a. Therefore, your domain would be “all real numbers x not equal to a”.
HORIZONTAL ASYMPTOTES
*If f(x) has a HORIZONTAL ASYMPTOTE, it can be determined by the ratio of the
highest degree terms in p(x) and q(x). This asymptote is a guiding line for large values of the
function (big positive x’s and big negative x’s).
To find the Horizontal Asymptote, make a fraction with only the highest degree term in p(x)
in the numerator and the highest degree term in q(x) as the denominator. Reduce this
fraction completely. If the fraction reduces to a number, then y = that number is your
horizontal asymptote equation. If the fraction reduces to number , then y = 0 is your
x
horizontal asymptote equation. [Remember the degree of a polynomial is the highest power
of the polynomial]
*f(x) will have a HORIZONTAL ASYMPTOTE only if the degree of q(x) degree of p(x).
NOTE: The information above is VERY general and probably very confusing at this
point since no examples were provided. Go carefully through the next two examples and
the information will make more sense.
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