Pre-CalculusChapter 2

Polynomial and Rational Functions

2

2.6 Rational Functions and Asymptotes

Objectives: Find the domains of rational functions. Find the horizontal and vertical

asymptotes of graphs of rational functions. Use rational functions to model and solve

real-life problems.

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Rational Functions A rational function can be written in the

form

where N(x) and D(x) are polynomials.

A rational function is not defined at values of x for which D(x) = 0.

)(

)()(

xD

xNxf

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Reciprocal Function

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Asymptotes An asymptote is a boundary line that the

graph of a function approaches, but never touches or crosses.

The line x = a is a vertical asymptote of the graph of f if, as x approaches a from either the left or the right, f (x) approaches ∞ or –∞.

The line y = b is a horizontal asymptote of the graph of f if, as x approaches ∞ or –∞, f (x) approaches b.

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Examples The following graphs show horizontal and

vertical asymptotes of two rational functions.

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Finding Asymptotes Let f be a rational function:

Vertical Asymptotes: Occur when the denominator equals

zero. Simplify the function if possible. Set D(x) = 0 and solve for x.

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Horizontal Asymptotes The graph of f has at most one horizontal

asymptote determined by comparing the degrees of N(x) and D(x).

Let n be the degree of the numerator and m be the degree of the denominator.

Let an be the leading coefficient of the numerator and bn be the leading coefficient of the denominator.

If n < m HA: y = 0

If n = m

HA: y = an/bn

If n > m No HA

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Examples Find all HA and VA of each rational function.

1

2)(.2

13

2)(.1

2

2

2

x

xxg

x

xxf

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Example Find all HA and VA of the rational function.

6

2)(

2

2

xx

xxxf

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For a person with sensitive skin, the amount of time T, in hours, the person can be exposed to the sun with a minimal burning can be modeled by

where s is the Sunsor Scale reading (based on the level of intensity of UVB rays).

a. Find the amount of time a person with sensitive skin can be exposed to the sun with minimal burning when s = 10, s = 25, and s = 100.

b. If the model were valid for all s > 0, what would be the horizontal asymptote of this function, and what would it represent?

1200,8.2337.0

ss

sT

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Homework 2.6

Worksheet 2.6# 7 – 12 (matching), 15, 17, 19,

35, 39