Rational functions
Download
1 / 32

Rational Functions - PowerPoint PPT Presentation


  • 464 Views
  • Uploaded on
  • Presentation posted in: General

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha

Download Presentation

Rational Functions

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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

  • 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.


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


Basics

Multiplying fractions:

Adding fractions:

Simplifying fractions:


Simplifying fractions with variables


Rational Functions

Definition

A rational function can be written in the form

Where and are both polynomial

functions and


Examples

Rational function

Rational function

Not a rational function


Domain of a Rational Function

The domain of a rational function

is given by:

Examples

Domain:

Domain:


End Behavior

Let be a rational function. The line

is a horizontal asymptote (HA) if:


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:


HA Examples

HA:


HA Examples (Continued)

HA:

No Horizontal Asymptote


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.


HA Examples

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

HA:


HA Examples (Continued)

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


General end behavior

Let be a rational function and let

Then the end behavior of is the same

as the end behavior of:


End behavior example

Consider the function

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

Its end behavior is the same as


Vertical Asymptotes

Let be a rational function.

The line is a vertical asymptote (VA) if:


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


VA Example

Solve:

VA:

(Note that is not a vertical asymptote!)


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


Example

Find the zeros of

1. Reduce the function to lowest terms.

2. Set the numerator equal to zero and solve


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):


Graphing a Rational Function (Cont’d)

6. Create a table for

Not in the domain! (open circle)


Graphing a Rational Function (Cont’d)


Solving a Rational Equation

Solve

Multiply both sides by

On the TI83/84 calculator:


Solving a Rational Equation

Solve

Multiply both sides by

No solution


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


Range of a Rational Function

1. Read from graph, or

2. Use the fact that:


Range of a Rational Function

Find the range of

Previously we found that

Domain of :

Range of :


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:


Questions?


ad
  • Login