0-3: Rational Functions & Asymptotes Objectives: Determine horizontal, vertical & slant asymptotes Graph rational functions ©2002 Roy L. Gover Modified by Mike Efram 2004 This lesson is not in your text

Definitions Rational Function: the ratio of two functions of the form:

Example g(x) h(x)

More Examples

Definitions Degree of a Function:The largest exponent in the function. Leading Coefficient: the coefficient of the term with the largest exponent.

Example 1. What is the degree of f(x)=x+2x 4 -3? 2. What is the leading coefficient of f(x) ?

Definition The line x=a is a vertical asymptote for f(x) if f(x) as x a from the left or from the right

What is true about f(x) when x = -2 ? Example

Try This Find all vertical asymptotes for: x=2,-3

Example Find all vertical asymptotes of: Hint: Factor to find where the denominator = 0!!

Definition The line y=b is a horizontal asymptote of f(x) if f(x) b as x

Example y =-3 is a horizontal asymptote

Rules for Finding Horizontal Asymptotes 1. If degree of numerator < degree of denominator, horizontal asymptote is the line y =0 ( x axis)

Rules for Finding Horizontal Asymptotes (cont.) 2. If degree of numerator =degree of denominator, horizontal asymptote is the line y =ratio of leading coefficients.

Rules for Finding Horizontal Asymptotes (cont.) 3. If degree of numerator >degree of denominator, there is no horizontal asymptote.

Example Find the horizontal asymptotes, if any, of: Graph and confirm

Example Find the horizontal asymptotes, if any, of: Graph and confirm

Try This Find the horizontal asymptotes, if any, of: Graph and confirm

Definition The slant line y=mx+b is a slant asymptote of f(x) if the degree of the numerator is exactly one greater than the degree of the denominator.

Guidelines for Finding Slant Asymptotes 1. Divide denominator into numerator using long division. Ignore any remainder. 2. Slant asymptote is y =the result of the above division.

Long Division of Polynomials (Needed to find slant asymptotes) Examples

Try This Divide using long division: Remainder

Example Find the slant asymptotes, if any for: Graph and confirm

Try This Find the slant asymptotes, if any for: Graph and confirm y=x+3

Example Consider the rational function 1. Find all asymptotes. 2. What’s happening at f(-1) ?

Try This Consider the rational function 1. Find all asymptotes. 2. What’s happening at f(0) ? 3. Graph (don’t use a calculator)

Example Consider the rational function: What is different? Hint: factor numerator & simplify.

Some rational functions are discontinuous... asymptote rules don’t apply. How do you know? Important Idea