1

Algebra 2: Section 6.2

Evaluating and Graphing Polynomial Functions

2

Polynomial Function

• A function is a polynomial function if…

– Exponents are all whole numbers

– Coefficients are all real numbers

• Standard Form of Polynomial Function

– All terms are written in descending order of

exponents from left to right

1

1 1 0( ) n n

n nf x a x a x a x a

3

Parts of Polynomial

Function

• Leading coefficient

– Coefficient on highest power of x

• Constant term

– Term that has no variable (no x)

• Degree of the polynomial

– Exponent of the highest power of x

4

Classifying Polynomial

Functions

• Classify based on highest power of x

• Power of x

– Zero: Constant

– One: Linear

– Two: Quadratic

– Three: Cubic

– Four: Quartic

5

Examples

2 21. ( ) 2f x x x

No, because negative exponent.

• Decide whether the function is a

polynomial function. If it is, write the

function in standard form and state its

degree, type, and leading coefficient.

6

Examples

3 42. ( ) 0.8 5g x x x

Yes4 3( ) 0.8 5f x x x

Degree: 4

Type: Quartic

Leading Coefficient: 1

7

Synthetic Substitution

(Synthetic Division)

• Gives another way to evaluate a function

• Also used to divide polynomials

– This will be discussed in later sections

• The last entry is the value of the function

8

Examples

• Use synthetic division to evaluate.

5 43. ( ) 3 5 10 when 2f x x x x x

9

5 43. ( ) 3 5 10 when 2f x x x x x

3 1 0 0 5 102

3

6

7

14

14

28

28

56

51

102

92

Coefficients of x written in order

Missing power of

x, zero coefficient!

Number you are

evaluating goes in

front

Drop 1st

number

down

10

3 24. ( ) 5 4 1; (4)f x x x x f

5 1 4 14

5

20

21

84

80

320

321

11

Assignment

• p.333

#15-26 all, 37-46 all

(22 problems)

What happens to the graph when x gets

very small or x gets very large?

12

As x gets very small

the graphs is falling

As x gets very large

the graphs is rising

What happens to the graph when x gets

very small or x gets very large?

13

As x gets very small

the graphs is rising

As x gets very large

the graphs is rising

14

End Behavior of Graphs of

Polynomial Functions

• What the function’s graph does as x approaches positive

and negative infinity

x

( )f x

x“as x approaches negative infinity”

OR as x gets very small

OR as we move forever to the left

“as x approaches positive infinity”

OR as x gets very large

OR as we move forever to the right

( )f x“f(x) approaches negative infinity”

OR y gets very small

OR the graph falls

“f(x) approaches positive infinity”

OR y gets very large

OR the graph rises

15

End Behavior of Polynomial Functions

1

1 1 0( ) ...n n

n nf x a x a x a x a

• Leading term and degree tell end

behavior

• Follow these rules…

16

End Behavior of Polynomial Functions

1

1 1 0( ) ...n n

n nf x a x a x a x a

0 and even,

( ) as and ( ) as

na n

f x x f x x

0 and even,

( ) as and ( ) as

na n

f x x f x x

17

End Behavior of Polynomial Functions

1

1 1 0( ) ...n n

n nf x a x a x a x a

0 and odd,

( ) as and ( ) as

na n

f x x f x x

0 and odd,

( ) as and ( ) as

na n

f x x f x x

18

End Behavior of Polynomial

Functions

Leading

Coefficient

Degree Left

Behavior

Right

Behavior

+ Even Rises Rises

+ Odd Falls Rises

- Even Falls Falls

- Odd Rises Falls

19

End Behavior of Polynomial

Functions

Leading

Coefficient

Degree

+ Even

+ Odd

- Even

- Odd

as x

( )f x ( )f x

( )f x( )f x

( )f x ( )f x

( )f x ( )f x

as x

20

Examples

• Describe the end behavior of the

function.3 21. ( ) 2 3f x x x x

Leading Coefficient: + Degree: Odd

( ) as

( ) as

f x x

f x x

21

Assignment

• p.334

#53-64 all, 65-79 odds

(20 problems)

• #65-79 odds (draw sketches of graphs

using graphing calculator, trace the curve

to get fairly accurate graphs)