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Rational Functions. Macon State College Gaston Brouwer, Ph.D. June 2010. Georgia Performance Standards. Mathematics 4. MM4A1. Students will explore rational functions. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of

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Rational functions l.jpg

Rational Functions

Macon State College

Gaston Brouwer, Ph.D.

June 2010


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Georgia Performance Standards

Mathematics 4

MM4A1. Students will explore rational functions.

  • Investigate and explain characteristics of rational

  • functions, including domain, range, zeros, points of

  • discontinuity, intervals of increase and decrease, rates

  • of change, local and absolute extrema, symmetry,

  • asymptotes, and end behavior.

b. Find inverses of rational functions, discussing domain

and range, symmetry, and function composition.

c. Solve rational equations and inequalities analytically,

graphically, and by using the appropriate technology.


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Rational Functions

  • Basics

  • What is a Rational Function?

  • Domain

  • Horizontal & Vertical Asymptotes

  • Zeros

  • Graphing a Rational Function

  • Solving Rational Equations

  • Inverses

  • Range

  • Solving Rational Inequalities


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Basics

Multiplying fractions:

Adding fractions:

Simplifying fractions:



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Rational Functions

Definition

A rational function can be written in the form

Where and are both polynomial

functions and


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Examples

Rational function

Rational function

Not a rational function


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Domain of a Rational Function

The domain of a rational function

is given by:

Examples

Domain:

Domain:


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End Behavior

Let be a rational function. The line

is a horizontal asymptote (HA) if:


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How to find a horizontal asymptote (1)

1. Divide and by the highest power of that

shows up in . Call the resulting functions and

.

2. HA:



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HA Examples (Continued)

HA:

No Horizontal Asymptote


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How to find a horizontal asymptote (2)

  • If degree( ) < degree( ), the HA is given by

2. If degree( ) = degree( ), the HA is given by

3. If degree( ) > degree( ), there is no HA.


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HA Examples

Degree( ) = degree( )=2, so:

HA:


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HA Examples (Continued)

Degree( ) > degree( ), so there is no HA.


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General end behavior

Let be a rational function and let

Then the end behavior of is the same

as the end behavior of:


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End behavior example

Consider the function

Degree( ) > degree( ), so there is no HA.

Its end behavior is the same as


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Vertical Asymptotes

Let be a rational function.

The line is a vertical asymptote (VA) if:


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How to find vertical asymptotes

1. Reduce the function to lowest terms.

2. The vertical asymptote(s) is (are):

where is (are) the solution(s) to


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VA Example

Solve:

VA:

(Note that is not a vertical asymptote!)


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How to find zeros of a rational function

1. Reduce the function to lowest terms.

2. The zeros of the rational function are the

solutions to


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Example

Find the zeros of

1. Reduce the function to lowest terms.

2. Set the numerator equal to zero and solve


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Graphing a Rational Function

Graph:

1. Reduce to lowest terms:

2. Find y-intercepts (set x=0):

3. Find zeros/x-intercepts (solve f(x)=0):

4. Find the horizontal asymptote:

5. Find the vertical asymptote(s):


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Graphing a Rational Function (Cont’d)

6. Create a table for

Not in the domain! (open circle)



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Solving a Rational Equation

Solve

Multiply both sides by

On the TI83/84 calculator:


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Solving a Rational Equation

Solve

Multiply both sides by

No solution


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Inverses

Find the inverse of

1. Write the function in the form y=…

2. Interchange x and y

3. Solve for y

3. Write in the form


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Range of a Rational Function

1. Read from graph, or

2. Use the fact that:


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Range of a Rational Function

Find the range of

Previously we found that

Domain of :

Range of :


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Solving a Rational Inequality

Solve

Write the equation in the form:

On a number line, mark all the points where

with a “0” and all the points where with a “?”.

Then determine the sign of

on each interval

by using test points.

On the TI83/84:



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