Question

Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station.

The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task.

(1, 4), (2, 8), (3, 16), (4, 32)

Part A: Is this data modeling a linear function or an exponential function? Explain your answer. (2 points)

Part B: Write a function to represent the data. Show your work. (4 points)

Part C: Determine the average rate of change between station 2 and station 4. Show your work. (4 points)

do not copy and paste from a different question because those aren't right.

Answer 1

A) Exponential function

`\textsf{B)} \quad y = 2(2)^x`

C) 12

Explanation:

Given points:

• (1, 4)
• (2, 8)
• (3, 16)
• (4, 32)

Part A

In a linear function, the y-values have equal differences.

In an exponential function, the y-values have equal ratios.

From inspection of the given points, we can see that y doubles each time x increases by 1 unit, therefore the data models an exponential function.

Part B

`\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

As the function doubles, the base is 2.

Substitute one of the points (1, 4) and b = 2 into the formula and solve for a:

`\implies 4=a \cdot 2^1`

`\implies 4=2a`

`\implies a = 2`

Substitute the found values of a and b into the formula to create the exponential function to represent the given data:

`y=2(2)^x`

Part C

`\boxed{\begin{minipage}{6.3 cm}\underline{Average rate of change of function $f(x)$}\\\\$(f(b)-f(a))/(b-a)$\\\\over the interval $a \leq x \leq b$\\\end{minipage}}`

Station 2: (2, 8)

Station 4: (4, 32)

To determine the average rate of change between station 2 and station 4 find the average rate of change over the interval 2 ≤ x ≤ 4.

Therefore, a = 2 and b = 4:

`\implies \textsf{Rate of change}=(f(4)-f(2))/(4-2)=(32-8)/(4-2)=(24)/(2)=12`