8 Pull It All Together

Image Long Description

BIG idea Proportionality

Inverse proportionality involves a relationship in which the products of two quantities remain constant as the corresponding values of the quantities change.

Task 1

Rectangle R has varying length ℓ and width w but a constant perimeter of 4 ft.

1. Express the area A as a function of ℓ. What do you know about this function?
2. For what values of ℓ and w will the area of R be greatest? Give an algebraic argument. Give a geometric argument.

Task 2

Rectangle R has varying length ℓ and width w but a constant area of 4 ft 2 . 4 , ft squared , .

1. Express the perimeter P as a function of ℓ. What kind of function is P? What is its domain?
2. Describe the asymptotic behavior of P. What can you say about R because of this behavior? Could you have made a similar statement about R in Task 1?
3. For what values of ℓ and w will the perimeter of R be least? Give a calculator-based argument. Give a geometric argument.

BIG idea Function

You can represent functions in a variety of ways (such as graphs, tables, equations, or words). Each representation is particularly useful in certain situations.

Task 3

Describe the discontinuities of f ( x ) = x 2 + x − 6 x 2 − 5 x + 6 . f , open x close , equals . fraction x squared , plus x minus 6 , over x squared , minus 5 x plus 6 end fraction . .  Find an equivalent form for f that shows how the graph of f is related to the graph of y = 1 x . y equals , 1 over x , .  Describe the relationship. (You do not have to draw the graphs, but you can if you wish.)

BIG idea Equivalence

You can use symbols to represent an equation in an unlimited number of ways, where all equations have the same solution.

Task 4

Solve x − 1 = x 4 − 2 x 3 x 2 − 4 x minus 1 equals . square root of fraction x to the fourth , minus 2 , x cubed , over x squared , minus 4 end fraction end root .

Table of Contents