Written Homework #9: Polynomial Functions

Question

(Standards 8 & 9)

1. (Standards 8 & 9) Given the graph below, write the equation of a

polynomial with the same end-behavior, x-intercepts, y-intercept

(0,-2) and multiplicities. Explain the reasoning behind your equation

using complete sentences.

2. (Standards 8 & 9) Let g(x) =— 2(x − 10)²(x − 1)(x + 5)³

a. Find the leading term for the polynomial g(x)

b. Explain how the leading term tells us what happens at the ends

of the graph.

c. Write out the end behavior using limits.

d. Find all of the x-intercepts and the y-intercept

e. Explain the behavior of g(x) near each x-intercept. Please

make sure to include how the behavior is different for each

x-intercept.

3. (Standards 8 & 9) Exploration: Zoey is a tile specialist at a local

pool company Hayward Pool and Tile. When an in-ground pool is

installed, she comes in and creates the border surrounding the edge

of the pool. She uses 1 foot long square tiles and completely

surrounds the pool, an example is shown below:

a. The pool in the picture above is a 2 X 2 pool and requires 12

tiles to create the border. Determine how many tiles are

needed to create the border for a 3 X 3 pool, a 4 X 4 pool, ... up

to a 7 X 7 pool. (Hint: draw a picture of the pool and the border)

b. Using the data you gathered above do you notice any patterns?

Based on that pattern, can you quickly determine how many

tiles it would take to create the border for an 8 X 8 pool? 9 X 9

pool? Describe using complete sentences how you would do

that.

c. Let n denote a positive integer. Using the work you did before,

write down an equation for the number of tiles needed to create

the border for an n X n pool. Mathematicians would say that

you "formulated a conjecture" in this step. Congratulations on

your new conjecture!!

d. Use the equation you came up with in part c to answer the

following: What is the size of the biggest pool Zoey can tile

around using 100 tiles?