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

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)

Answer 1

Part A:

The data does not follow a linear pattern since the differences between the y-coordinates are not constant. However, the data follows an exponential pattern, as each y-coordinate is double the previous one.

Part B:

We can write the function as f(x) = ab^x, where a is the initial time and b is the growth factor. Using the given data, we can find the values of a and b as follows:

f(1) = ab^1 = 2, so a = 2/b

f(2) = ab^2 = 4, so 4 = (2/b)b^2, which simplifies to b = 2

Thus, a = 1 and the function is f(x) = 2^x.

Part C:

The average rate of change between station 2 and station 4 is equal to the slope of the line passing through those two points. Using the coordinates (2, 4) and (4, 16), we can calculate the slope as:

slope = (16 - 4) / (4 - 2) = 6

Therefore, the average rate of change between station 2 and station 4 is 6 minutes per station.