Question

part 4 of 5
the number of teams y remaining in a single elimination tournament can be found using the exponential function $y = 128\left(\frac{1}{2}\
ight)^x$, where $x$ is the number of rounds
a through d below.

a. determine whether the function represents exponential growth or decay. explain
the function represents exponential decay because the value of b is less than 1

b. what does 128 represent in the function? choose the correct answer below
a. the initial number of teams in the tournament
b. the growth rate for the number of teams in the tournament
c. the decay rate for the number of teams in the tournament
d. the number of games played in the tournament

c. what percent of the teams are eliminated after each round? explain how you know.
the rate of decrease is $\frac{1}{2}$, so 50 % of the teams are eliminated after each round

d. what is a reasonable domain and range for the function? explain
a reasonable domain is $0\leq x\leq7$ and a reasonable range is $1\leq y\leq128$ because the function is an exponential function.

Explanation:

Step1: Identify growth/decay

For $y=ab^x$, if $0

Step2: Interpret initial value

In $y=ab^x$, $a$ is the starting value. So 128 = initial team count.

Step3: Calculate elimination percent

Decay factor $b=\frac{1}{2}$, so remaining % is $50\%$. Eliminated: $100\%-50\%=50\%$.

Step4: Find domain/range

Domain: $x$ is rounds. $y=1$ when $x=7$ ($128(\frac{1}{2})^7=1$), so $0\leq x\leq7$. Range: Starts at 128, ends at 1, so $1\leq y\leq128$.

Answer:

a. Exponential decay
b. A. The initial number of teams in the tournament
c. 50%
d. Domain: $0 \leq x \leq 7$; Range: $1 \leq y \leq 128$