BELLWORK1. Given the following function, a) determine the direction of its

opening b) find the axis of symmetry c) the vertex

13

4

3

2)( 2 xxxf

2. A track and field playing area is in the shape of a rectangle with semicircles at each end. The inside perimeter of the track is to be 1500 meters. What should be the dimensions of the rectangle so that the area of the rectangle is a maximum?

Pre-Calculus Honors Day 15

2.2 Polynomial Functions of High Degrees- How do you sketch graphs of polynomial function?

-How to determine end behavior of graphs of polynomial functions?-How to find the zeros of polynomial functions?

012

21

1 ...)( axaxaxaxaxf nn

nn

965432 xyxyxyxyxyxy

•Polynomial function of x with degree n…

Graphs of Polynomial Functions are continuous with no breaks, holes, or gaps. The have smooth rounded turns with no sharp pointed turns.

Even Functions: If n is even, the graph of y = xn touches the axis at the x-intercept.

Odd Functions: If n is odd, the graph of y = xn crosses the axis at the x-intercept.

Compare the following functions: Sort the functions by any method (s) you choose.

The Leading Coefficient Test

• When n is odd• If the leading coefficient is positive (an > 0), the graph falls to

the left and rises to the right.

• If the leading coefficient is negative (an < 0), the graph rises to the left and falls to the right.

• When n is even • If the leading coefficient is positive (an > 0), the graph rises to

the left and right.

• If the leading coefficient is negative (an < 0), the graph falls to the left and right.

012

21

1 ...)( axaxaxaxaxf nn

nn

Example1: Use the leading coefficient test to determine the left and right behavior of the graph of each polynomial function.

xxxfa 4)() 3

45)() 24 xxxfb

xxxfc 5)()

Degree = 3 (odd), LC = negativeRises to the left and falls to the right

Degree = 4 (even), LC = positiveRises to the left and right

Degree = 5 (odd), LC = positiveFalls to the left and rises to the right

What’s going on in the middle?

• USE THE ROOTS!• Single Root (x - c): simply crosses at x = c.• Double Root (x – c)2: graph touches but does not cross at x = c. Graph “bounces” at c.

• Triple Root (x – c)3: graph will flatten at x = c as it passes the x-axis.

Example 2: sketch the graph using the leading coefficient test and the roots.

3)3)(4(4)() xxxfa

2)3(2)() xxbb

)1)(1(3)() 2 xxxhc

)1)(2)(3() xxxyd

)2)(3)(1)(1() xxxxye

2)3)(1() xxyf

left rises, right rises, crosses at -4, flattens at -3

Left falls, right falls, bounces at -3

Left rises, right falls, crosses at -1

left rises, right falls, crosses at -3, -2, 1

Left rises, right rises, crosses at -1, 1, -3, 2

Left falls, right rises, crosses at 1, bounces at 3

Example 3: Find the real zeros and then graph.

xxxxfa 2)() 23 24 22)() xxxfb

)1)(2(

)2(

202

23

xxx

xxx

xxx

)1)(1(2

)1(2

220

2

22

24

xxx

xx

xx

Left falls, right rises, Crosses x-axis at 0, 2, -1

Left falls, right fallsCrosses x-axis at 1, -1Double root at x = 0 bounces

)43(

4303

34

xx

xx

Example 3: Find the real zeros and then graph.

34 43)() xxxfc xxxxfd2

962)() 23

2

2

23

)32(2

1

)32)(32(2

1

)9124(2

12

9620

xx

xxx

xxx

xxx

Left rises, right rises, Crosses at 4/3, and Flattens as it goes through 0. Left rises, right falls,

Crosses at 0, bounces at 3/2

• P. 156• 20, 22, 31, 32, 56, 59, 72

Polynomial Art!

• Graph a total of three polynomial functions (at least one even- and one odd-degree function) that you come up with yourself. Color each intersecting region differently so that no two bordering regions are colored alike. Write the equations of the functions on a key on the poster. • Title: 10%• At least three functions: 20% each = 60%• Neatness and Originality: 30%

Tonight's Homework

Pg 156

#19,21, 23, 27, 30, 34, 55, 57

Polynomial Art Due Next Friday 2/22