Notes Over 3.2Graphs of Polynomial FunctionsGraphs of Polynomial Functions

Continuous Functions Non-Continuous Functions

Polynomial functions are continuous

Notes Over 3.2Graphs of Polynomial FunctionsGraphs of Polynomial Functions

Rounded Turns Sharp Turns

Polynomial functions have rounded turns

Notes Over 3.2Graphs of Polynomial FunctionsGraphs of Polynomial Functions

Simplest Graphs of the Form:

When n is odd

nxy

When n is even

5xy 3xy

4xy 2xy

Notes Over 3.2Sketching Transformations of Monomial FunctionsSketching Transformations of Monomial Functions

42 .1 xy

Left 2

Down

Notes Over 3.2Sketching Transformations of Monomial FunctionsSketching Transformations of Monomial Functions

1 .2 5 xy

Down 1

Upwards

Notes Over 3.2Leading Coefficient and Degree TestLeading Coefficient and Degree Test

If leading coefficient is positive, it ends going up.If leading coefficient is negative, it ends going downIf degree is odd, it starts and ends on opposites sidesIf degree is even, it starts and ends on the same side

523 23 xxxy 4

Notes Over 3.2Leading Coefficient and Degree TestLeading Coefficient and Degree Test

If leading coefficient is positive, it ends going up.If leading coefficient is negative, it ends going downIf degree is odd, it starts and ends on opposites sidesIf degree is even, it starts and ends on the same side

241.2 .3 35 xxyDetermine right-hand and left-hand behavior of each.

Because the leading coefficient is negative – –

Because the degree is odd (the opposite)– Because the degree is odd (the opposite)–

it falls to the right.it falls to the right.it falls to the right.it falls to the right.

it rises to the left.it rises to the left.it rises to the left.it rises to the left.

Notes Over 3.2Leading Coefficient and Degree TestLeading Coefficient and Degree Test

If leading coefficient is positive, it ends going up.If leading coefficient is negative, it ends going downIf degree is odd, it starts and ends on opposites sidesIf degree is even, it starts and ends on the same side

5

7514 .4

24

xx

y

Determine right-hand and left-hand behavior of each.

Because the leading coefficient is positive – –

Because the degree is even (the same)– Because the degree is even (the same)–

it rises to the right.it rises to the right.it rises to the right.it rises to the right.

it rises to the left.it rises to the left.it rises to the left.it rises to the left.

Goes straight through

Bounces off point

Notes Over 3.2Zeros of a Polynomial FunctionZeros of a Polynomial Function

Let n be the degree of the function.Then (n – 1) is the most turns the graph will have.n is the most number of zeros of the function (x-int).

24 33

1 .5 xxy

Find all of the zeros of the function, and use it to determine the number of turning points.

0 2

3

1x 2x 9

3

1 2x x 3 0 x 3

x 3 x 3x 0 Three turning pointsThree turning pointsThree turning pointsThree turning points

Goes through with curve

Goes straight through

Notes Over 3.2Sketch the graph of the Polynomial FunctionSketch the graph of the Polynomial Function

43 2 5 .6 xxxf

0 3x x2 5

2

5x0x

Three turning pointsThree turning pointsThree turning pointsThree turning points

052 34 xx

it falls to the right.it falls to the right.it falls to the right.it falls to the right.

it falls to the left.it falls to the left.it falls to the left.it falls to the left. 1 1 2 2

Goes straight through

Bounces off

Notes Over 3.2Sketch the graph of the Polynomial FunctionSketch the graph of the Polynomial Function

xxxxf 20 20 5 .7 23 0 x5 2x x4

2x0x

Two turning pointsTwo turning pointsTwo turning pointsTwo turning points

it rises to the right.it rises to the right.it rises to the right.it rises to the right.

it falls to the left.it falls to the left.it falls to the left.it falls to the left.

1 1 1

4 0 5 x x x2 2

025 2 xx

Notes Over 3.2Writing Polynomial FunctionWriting Polynomial FunctionFind a polynomial with degree 4 that has the given zero.

2 ,1 ,3 : Zeros.8 0 23x 1x 2x 0 2x x2 3 2x x4 4

4x 34x 24x32x 28x x8

23x x12 12 4xxf 32x 27x x20 12

Notes Over 3.2