Determine the following algebraically (no calculator)

a) Vertex

b) x-intercepts

c) y- intercepts

d) Is the vertex a max or min? How do you know without

graphing?

1123)( 1. 2 xxxfQuadratics

Section 3.2 Polynomial Functions and Models

1. Polynomial Functions and their degree

Find the degree (largest power when written in standard form).

Factored form: Add the degrees contributed by each factor523 )2()2()( :2 xxxxfExample

32)( :1 6 xxxfExample

f (x) anxn an 1x

n 1 ... a1x a0

A polynomial function is a function of the form

Coefficients are: n is non-negative integer.

2. Properties of the Power Functions

A power function is of the form

f (x) x n

If n is odd integer If n is even integer

Symmetry:

Domain:

Range:

3. End Behavior of Polynomials

Given a polynomialAs x increases/decreases without bound, the highest degree term determines the end behavior of f(x).

01)( axaxaxaxf nn

nn

nnxa

3. End Behavior of the graph

Even Degree and

Leading Term Testlook at leading term (from standard form)

nnxa

Leading Coefficient is Positive

,..., 42 xx

Leading coeffic.

is Negative,..., 42 xx

Odd Degree and

Leading coefficient is Positive

,..., 53 xx

Leading coefficient is Negative

,..., 53 xx

3b) Determine End Behavior using Leading Coefficient Test

1)

2)

3)

112)( xxxf

nnxa

xxxf 23)(

322 )2()2(2.0)( xxxxf

p. 183 #23-34. What is the end behavior of these graphs?

3c) End Behavior can also be used to determine Window Size on Calculator

248)( 234 xxxxf

How do you know if you are viewing the

entire graph?

Set window to [-5,5] x [-5,5] and graph:

41013)( 23 xxxxf

4. Zeroes of Polynomials and their Multiplicity

Factor Theorem If is a polynomial function, for which

the following are equivalent statements:

(1) r is a zero or root of

(2) r is an x-intercept of

(3) is a factor of

f

f

f

f)( rx

0)( rf

Example 1: Determine all zeros of f.82)( 2 xxxf

4 Zeroes of Polynomials and their MultiplicityDefinition: The multiplicity of a zero is the degree of the factor

Example: If f has a factor , then 3 is a zero with multiplicity 4

4)3( x

Example:

Determine: The zeros of f and multiplicity of each :

)2()3(2)( 42 xxxxf

Multiplicity greater than 1 represents a repeated zero

5. Investigating the Role of Multiplicity.How does Multiplicity affect the behavior of the graph at the zero?

Graph each function, what do you notice changes at the zero when the degree is even or odd?

1) 3)

2) 4)

Does the sign of change on each side of the zero?)(xf

3)3()( xxf3)4()( xxf

2)3()( xxf2)4()( xxf

6. Behavior of Graph at a ZeroMultiplicity tells us the behavior of the graph at a zero (x-intercept).

If multiplicity , m, is a number that is:

Behavior of graph at the zero

Even touches (is tangent)Odd crosses (changes sign)

6. Example Continued

Zeros Multiplicity Is m an even or odd integer?

Behavior of graph at the zero

0 2 3 4 -2 1

Previous Example . )2()3(2)( 42 xxxxf

6. More Practice

342 )1()1()( xxxxf

State the degree of this polynomial.How many zeros does this function have?

Zeros of the function

Multiplicity of the zero

Is meven or odd?

Shape of the graph near the zero

Use your graphing calculator to graph the following:1) 2)2()( xxf 6)2()( xxf

7. Large values of Multiplicity

Analyze the graph of

What happens at the zero as m gets large?

mxxf )2()(

2)3)2()( xxf 7)2()( xxf

The graph flattens out at the zero as the multiplicity increases.

8. Graphing a Polynomial Function

Given the polynomial

1. Leading Term: 2. End behavior:

3. y-intercept:

4. x-intercepts (i.e. the zeros)

23 )3)(2()1()( xxxxf

Zeros Multiplicity even or odd? Cross/Touch

8. Graphing a Polynomial Function

Given the polynomial

Before finding zeros: Write in completely factored form1. Leading Term:2. End behavior

3. y-intercept4. x-intercepts (i.e. the zeros)

)4(5.)( 23 xxxf

Zeros Multiplicity even or odd? Cross/Touch

8. Graphing a Polynomial Function

Your turn...

1. Degree2. End behavior

3. y-intercept4. x-intercepts (i.e. the zeros)

52 )5)(1()1()( xxxxxf

Zeros Multiplicity even or odd? Cross/Touch

1. End behavior

2. y-intercept

3. x-intercepts (i.e. the zeros)

8. Graphing a Polynomial Function

When polynomial is not in factored form, find zeros using:

Factoring or Graphing Calculator.

9182)( 23 xxxxf

Zeros Multiplicity even or odd? Cross/Touch

10. Using the graphing calculator

Graph: p. 184 #81.

x-intercepts: Use ZERO feature y-intercepts: TRACE: x=0

d) Table to determine graph close to zero. Is it above or below?

e) Max/Min

Find zeros (x-intercepts) using graphing calculator.

9. a) Building Polynomials from zerosGiven that f has zeros with multiplicitywe can write:

Write one possible polynomial with these properties:1) Degree 4: Zeroes at: -1, 2, 5 ;

21 )()()( 21mm rxrxaxf

,, 21 rr ,, 21 mm

2) x-intercepts at: (-3,0), (4,0), (-1,0), (0,0) and the graph rises to the left and falls to the right.

Polynomial – turning points

Definition: turning point - graph changes between increasing and decreasing.

If polynomial is degree n

Then it will have AT MOST n-1 turning points.

9. a)Building Polynomials from a graph

Use y-intercept to find scale factor.

Zerosx-intercepts

C/T ? Multiplicity

More practice….Construct a polynomial function that might have this graph.

Use higher-degree if graph is “flat” at the zero

Zerosx-intercepts

C/T ? Multiplicity

11. Finding a function of best fitUsing the graphing calculator, we can find a function of best fit for the following relationships:

Quadratic :cbxaxy 2 Cubic:

dcxbxaxy 23

Count the # of turning points

to determine best function

9. a)Building Polynomials from a graph

12. Analyzing the Graph of a Polynomial Function

Given the polynomial

1. End behavior.

2. x-intercepts. Solve f(x) = 0

)1)(3()4(2)( 22 xxxxxf

3. y-intercepts. Find f(0).

a) Behavior at each intercept (even/odd)

b) If k > 1, graph flattens for larger values of k.

5. Turning points: Graph changes between increasing/decreasing.

4. Symmetry: Odd/Even