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