Polynomial and Rational FunctionsLesson 2.3

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Note howmathematicsare referencedin the creation of cartoons

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We need a wayto take a numberof pointsand makea smoothcurve

This lesson studies

polynomials

This lesson studies

polynomials

Polynomials

General polynomial formula

• a0, a1, … ,an are constant coefficients

• n is the degree of the polynomial• Standard form is for descending powers of x

• anxn is said to be the “leading term”

Note that each term is a power function

11 1 0( ) ...n n

n nP x a x a x a x a

Family of Polynomials

Constant polynomial functions• f(x) = a

Linear polynomial functions• f(x) = m x + b

Quadratic polynomial functions• f(x) = a x2 + b x + c

Family of Polynomials

Cubic polynomial functions• f(x) = a x3 + b x2 + c x + d• Degree 3 polynomial

Quartic polynomial functions• f(x) = a x4 + b x3 + c x2+ d x + e• Degree 4 polynomial

Properties of Polynomial Functions

If the degree is n then it will have at most n – 1 turning points

End behavior• Even degree

• Odd degree

••

•

or

or

Properties of Polynomial Functions

Even degree• Leading coefficient positive

• Leading coefficient negative

Odd degree• Leading coefficient positive

• Leading coefficient negative

Rational Function: Definition

Consider a function which is the quotient of two polynomials

Example:

( )( )

( )

P xR x

Q x Both polynomials

2500 2( )

xr x

x

Long Run Behavior

Given

The long run (end) behavior is determined by the quotient of the leading terms• Leading term dominates for

large values of x for polynomial• Leading terms dominate for

the quotient for extreme x

11 1 0

11 1 0

...( )

...

n nn nm m

m m

a x a x a x aR x

b x b x b x b

nnm

m

a x

b x

Example

Given

Graph on calculator• Set window for -100 < x < 100, -5 < y < 5

2

2

3 8( )

5 2 1

x xr x

x x

Example

Note the value for a large x

How does this relate to the leading terms?

2

2

3

5

x

x

Try This One

Consider

Which terms dominate as x gets large

What happens to as x gets large?

Note:• Degree of denominator > degree numerator• Previous example they were equal

2

5( )

2 6

xr x

x

2

5

2

x

x

When Numerator Has Larger Degree

Try

As x gets large, r(x) also gets large

But it is asymptotic to the line

22 6( )

5

xr x

x

2

5y x

Summarize

Given a rational function with leading terms

When m = n• Horizontal asymptote at

When m > n• Horizontal asymptote at 0

When n – m = 1• Diagonal asymptote

nnm

m

a x

b x

a

b

ay x

b

Vertical Asymptotes

A vertical asymptote happens when the function R(x) is not defined• This happens when the

denominator is zero

Thus we look for the roots of the denominator

Where does this happen for r(x)?

( )( )

( )

P xR x

Q x

2

2

9( )

5 6

xr x

x x

Vertical Asymptotes

Finding the roots ofthe denominator

View the graphto verify

2 5 6 0

( 6)( 1) 0

6 or 1

x x

x x

x x

2

2

9( )

5 6

xr x

x x

Assignment

Lesson 2.3

Page 91

Exercises 3 – 59 EOO