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Intermediate Algebra – Student Workbook – Second Edition 2013

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Lesson 6a - Exponents and Rational Functions<br />

Mini-Lesson<br />

RATIONAL FUNCTIONS <strong>–</strong> KEY CHARACTERISTICS<br />

A RATIONAL FUNCTION is a function of the form<br />

f (x) = p(x)<br />

q(x)<br />

where p(x) and q(x) are polynomials and q(x) does not equal zero (remember that division by<br />

zero is undefined). Rational functions have similar shapes depending on the degree of the<br />

polynomials p(x) and q(x). However, the shapes are different enough that only the general<br />

characteristics are listed here and not a general graph:<br />

VERTICAL ASYMPTOTES<br />

f(x) will have VERTICAL ASYMPTOTES (can be more than one) at all input values where<br />

q(x) = 0. These asymptotes are vertical guiding lines for the graph of f(x) and f(x) will never<br />

cross over these lines.<br />

To find the Vertical Asymptotes for f(x), set q(x) = 0 and solve for x. For each x value that<br />

you find,<br />

x = that value is the equation of your vertical asymptote. Draw a dotted, vertical line on<br />

your graph that represents this equation.<br />

DOMAIN<br />

<br />

The x’s you found when solving q(x) = 0 are the values that are NOT part of the domain of<br />

f(x).<br />

To write your domain, suppose that your function denominator is x <strong>–</strong> a. When you solve x <strong>–</strong><br />

a = 0, you get x = a. Therefore, your domain would be “all real numbers x not equal to a”.<br />

HORIZONTAL ASYMPTOTES<br />

*If f(x) has a HORIZONTAL ASYMPTOTE, it can be determined by the ratio of the<br />

highest degree terms in p(x) and q(x). This asymptote is a guiding line for large values of the<br />

function (big positive x’s and big negative x’s).<br />

To find the Horizontal Asymptote, make a fraction with only the highest degree term in p(x)<br />

in the numerator and the highest degree term in q(x) as the denominator. Reduce this<br />

fraction completely. If the fraction reduces to a number, then y = that number is your<br />

horizontal asymptote equation. If the fraction reduces to number , then y = 0 is your<br />

x<br />

horizontal asymptote equation. [Remember the degree of a polynomial is the highest power<br />

of the polynomial]<br />

*f(x) will have a HORIZONTAL ASYMPTOTE only if the degree of q(x) degree of p(x).<br />

NOTE: The information above is VERY general and probably very confusing at this<br />

point since no examples were provided. Go carefully through the next two examples and<br />

the information will make more sense.<br />

Scottsdale Community College Page 246 <strong>Intermediate</strong> <strong>Algebra</strong>

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