Question

PLEASE HELP QUICKK 100 POINTS DUE TODAY!

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)

Answer 1

Part A: The given data is modeling an exponential function. In an exponential function, the independent variable (in this case, the station number) is raised to a constant power, resulting in an exponential growth or decay.

In this case, as the station number increases, the time taken to complete the task is doubling. For example, from station 1 to station 2, the time taken increases from 4 minutes to 8 minutes, which is a doubling of the time. Similarly, from station 2 to station 3, the time increases from 8 minutes to 16 minutes, again doubling the time. This pattern continues, indicating exponential growth.

Part B: To write a function representing the data, we can use the formula for exponential growth:

y = a * r^x

where:

y = time taken (in minutes)

x = station number

a = initial time taken at station 1

r = common ratio (rate of change)

Let's use the given data to find the values of a and r:

From the data, we have:

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

Substituting the values into the formula, we can solve for a and r:

Using (1, 4):

4 = a * r^1

Using (2, 8):

8 = a * r^2

Dividing the second equation by the first equation, we get:

2 = r^1

Therefore, r = 2.

Substituting r = 2 into the first equation, we get:

4 = a * 2^1

Simplifying, we find:

4 = 2a

Dividing both sides by 2, we get:

a = 2

So, the function representing the data is:

y = 2 * 2^x

Part C: To determine the average rate of change between station 2 and station 4, we need to find the slope of the line connecting the two points.

Using the coordinates (2, 8) and (4, 32), we can calculate the slope using the formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the values, we get:

slope = (32 - 8) / (4 - 2)

= 24 / 2

= 12

Therefore, the average rate of change between station 2 and station 4 is 12 minutes per station.

Explanation: