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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

Rational Functions

Macon State College

Gaston Brouwer, Ph.D.

June 2010

slide2

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 functions3
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
Basics

Multiplying fractions:

Adding fractions:

Simplifying fractions:

rational functions6
Rational Functions

Definition

A rational function can be written in the form

Where and are both polynomial

functions and

examples
Examples

Rational function

Rational function

Not a rational function

domain of a rational function
Domain of a Rational Function

The domain of a rational function

is given by:

Examples

Domain:

Domain:

end behavior
End Behavior

Let be a rational function. The line

is a horizontal asymptote (HA) if:

how to find a horizontal asymptote 1
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 continued
HA Examples (Continued)

HA:

No Horizontal Asymptote

how to find a horizontal asymptote 2
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 examples14
HA Examples

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

HA:

ha examples continued15
HA Examples (Continued)

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

general end behavior
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
End behavior example

Consider the function

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

Its end behavior is the same as

vertical asymptotes
Vertical Asymptotes

Let be a rational function.

The line is a vertical asymptote (VA) if:

how to find vertical asymptotes
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
VA Example

Solve:

VA:

(Note that is not a vertical asymptote!)

how to find zeros of a rational function
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
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
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
Graphing a Rational Function (Cont’d)

6. Create a table for

Not in the domain! (open circle)

solving a rational equation
Solving a Rational Equation

Solve

Multiply both sides by

On the TI83/84 calculator:

solving a rational equation27
Solving a Rational Equation

Solve

Multiply both sides by

No solution

inverses
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
Range of a Rational Function

1. Read from graph, or

2. Use the fact that:

range of a rational function30
Range of a Rational Function

Find the range of

Previously we found that

Domain of :

Range of :

solving a rational inequality
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: