The total cost in dollars to buy uniforms for the players on a volleyball team can be found using the function `c=30.75u+5.75,`where `u` is the number of uniforms bought. If there are at least 8 players but not more than 12 players on the volleyball team, what is the range of the function for this situation?

Question

There are 128 teams in a softball tournament. In each round, half of the teams are eliminated. Which function can be used to find the number of teams remaining in the tournament after `x` r

What is the value of the y-intercept of the graph of `f(x)=10.5(3.6)^{x}` ?

An investor deposited money into an investment account that earns interest compounded annually. The function shown models the amount of money in the account in dollars after `t` years.

`A\left(t\right)=1,550\left(1.02\right)^{t}`

Which statement best interprets one value of the function?

Convert the equation `y=-2\left(x-3\right)^{2}+1` from vertex form to standard form.

Answer 1

Let's address each question one by one:

1. Range of the function for uniform costs: The function given is \( c = 30.75u + 5.75 \). The number of uniforms \( u \) can range from 8 to 12 (inclusive). To find the total cost at these values of \( u \):

   For \( u = 8 \): \[ c = 30.75(8) + 5.75 = 246 + 5.75 = 251.75 \]

   For \( u = 12 \): \[ c = 30.75(12) + 5.75 = 369 + 5.75 = 374.75 \]

   Therefore, the range of the function \( c \) as \( u \) varies from 8 to 12 is: \[ [251.75, 374.75] \]

2. Function for teams remaining in a softball tournament: In the tournament, half of the teams are eliminated in each round. If there are initially \( n \) teams, the number remaining after \( x \) rounds can be expressed using the function: \[ f(x) = \frac{n}{2^x} \] If we start with 128 teams: \[ f(x) = \frac{128}{2^x} \]

3. Y-intercept of the function \( f(x)=10.5(3.6)^{x} \): The y-intercept occurs when \( x = 0 \). \[ f(0) = 10.5(3.6)^{0} = 10.5 \times 1 = 10.5 \]

4. Best interpretation of the investment function: The function modeling the amount of money in the account is: \[ A(t) = 1,550(1.02)^{t} \] This means that the initial amount deposited into the account is $1,550, and the amount increases by 2% each year. So, one interpretation can be: "After \( t \) years, the total amount in the investment account grows at an annual interest rate of 2%."

5. Convert the equation from vertex form to standard form: The vertex form given is: \[ y = -2(x-3)^{2} + 1 \] We can convert it to standard form \( y = ax^2 + bx + c \) by expanding: \[ y = -2((x-3)(x-3)) + 1 \] \[ = -2(x^2 - 6x + 9) + 1 \] \[ = -2x^2 + 12x - 18 + 1 \] \[ = -2x^2 + 12x - 17 \] Thus the function in standard form is: \[ y = -2x^2 + 12x - 17 \]