Graphing Polynomial Functions

By the end of today’s class, I will be able to: understand the significance of the zeros graph from factored form graph from “factorable” standard form

What do the characteristics indicate? Recall: The The degreedegree of the polynomial and of the polynomial and

and the and the signsign of the leading coefficient of the leading coefficientindicate the end-behaviour of the functionindicate the end-behaviour of the function

Consider: y = 2x3 + 4x2 - 38x - 40Since Degree = 3 is odd Therefore the graph has opposite ends Since Leading Coefficient = 2 is positive Therefore the graph generally rises right

E.B.(End Behaviour):As x -oo , y -oo

As x +oo, y +oo

Y-Intercept = -40

Possible Cubic Graphs# of

Turning Points# of Zeros

Zero Turning Points

2 Turning Points

1 zero

2 zeros

3 zeros

Y-Intercept = - 40

What do the zeros indicate?y = 2x3 + 4x2 - 38x - 40may be written asy = 2( x + 5 )( x + 1 )( x – 4 )

Since it is a cubic polynomial function with 3 zeros

Therefore the graph must have 2 turning points!

x-ints = - 5, -1, 4 y-int = - 40

-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

-90

-80

-70

-60

-50

-40

-30

-20

-10

10

20

30

40

50

60

x

y

What does the order of the roots indicate?Consider y = ( x – 5 )2

A “Double” Root means that the graph “Bounces”“Bounces” at that x-intercept!

===============================Consider y = ( x – 5 )3

A “Triple” Root means that the graph “Goes Through”“Goes Through” that x-intercept!

-2 -1 1 2 3 4 5 6 7 8 9

-2

-1

1

2

3

4

5

x

y

1 2 3 4 5 6 7

-2

-1

1

2

3

4

5

x

y

Now try these…#1] y = ( x – 2 )3 ( x + 5 )

#2] y = - 2/3 ( x – 3 )2 ( x + 1 )

#3] y = 3 ( 2x + 1 )2 ( 1 – x ) ( 3x - 5 )

#4] y = - x5 - 6x4 - 9x3

#5] y = 3x3 + 9x2 - 3x - 9

READ: p.139(right column) & p.140-144& “In Summary” on p.145

DO: p.146 #1,2b,6acef,10b,12b