7.1 Polynomial Functions7.1 Polynomial Functions

•Objective: •Evaluate Polynomials•Identify general shapes of graphs of polynomials

Polynomial FunctionPolynomial Function(in one variable)(in one variable)

degree of f(x): highest exponent

leading coefficient: the coefficient of the term with the highest degree

1 20 1 2 1( ) ...n n

n n nf x a x a x a x a x a

0a

0 0a

n

Examples:Examples:State the degree & leading

coefficient

1.

2.

3.

3 27 4z z z

3 2 26 4a a ab

2 49 3y y y

degree: 3

leading coeff.: 7

not a poly in 1 variable

degree: 4

leading coeff.: 1

Evaluate a Polynomial Evaluate a Polynomial FunctionFunction

Find f(6)

Find g(3)

2( ) 3 3 1f r r r 2(6) 3(6) 3(6) 1 37f

3( ) 2 4 ( )g x x x 3(3) 2 4(3) ( (3) )g

2 12 ( 27) 41

Find f(a2)

Find f(h+3)

2( ) 3 3 1f r r r

2 2 2 2( ) 3( ) 3( ) 1f a a a 4 23 3 1a a

2( 3) 3( 3) 3( 3) 1f h h h 23( 6 9) 3 9 1h h h

23 15 19h h

23 18 27 3 8h h h

Graphs of Polynomial Graphs of Polynomial FunctionsFunctionsConstant

Functiondegree = 0

=4

Linear Functiondegree = 1

4y 2 1y x

Graphs of Polynomial Graphs of Polynomial FunctionsFunctionsQuadratic

Functiondegree = 2

=4

Cubic Functiondegree = 3

2 3y x 312

2y x

Graphs of Polynomial Graphs of Polynomial FunctionsFunctionsQuartic Function

degree = 4

Quintic Functiondegree = 5

4 33.5 5 1y x x x 5 35 5 1y x x x

End Behavior of End Behavior of EVEN EVEN DEGREE DEGREE

Leading Coefficient Positive

Leading Coefficient Negative

( )

as

f x

x

( )

as

f x

x

( )

as

f x

x

( )

as

f x

x

End Behavior of End Behavior of ODD ODD DEGREE DEGREE

Leading Coefficient Positive

Leading Coefficient Negative

( )

as

f x

x

( )

as

f x

x

( )

as

f x

x

( )

as

f x

x

Zeros (or Roots)Zeros (or Roots)The number of REAL ZEROS a polynomial can be determined by how many times the graph crosses the x-axis.

For each graph, For each graph, - End Behavior?- End Behavior?- Even or Odd?- Even or Odd?

- # of - # of Real Real Zeros? Zeros?

Odd3 Real Zeros

Even1 Real Zeros

Odd3 Real Zeros