Question

A talent competition on television had five elimination rounds. After each elimination, only one-third of the contestants were sent to the next round. The table below shows the number of contestants in each round of the competition.

x12345
f(x)243812793


Compute the average rate of change of f(x) from x = 1 to x = 4 and describe what it represents.

Answer 1

Average rate of change is -78.

Explanation:

A talent competition on television had five elimination rounds. After each elimination, only one-third of the contestants were sent to the next round.

The table that shows the number of contestants in each round of the competition is,

x f(x)

1 243

2 81

3 27

4 9

5 3

Average rate of change is the change in the y-value divided by the change in the x-value for two points on the graph.

i.e
`(f(x_2)-f(x_1))/(x_2-x_1)`

So, the average rate of change of
`f(x)` from
`x = 1` to
`x = 4` is,

`=(f(4)-f(1))/(4-1)`

`=(9-243)/(4-1)`

`=(-234)/(3)`

`=-78`

Since f(x) is decreasing, so rate of change is negative.

Answer 2

for the answer to the question above,

Given the number of the contestants in each round:

X 1 2 3 4 5

F(x) 243 81 27 9 3

The rate of change can be calculated using y2-y1/x2-x1 formula. From rounds 1 to 4 the rate of change are

-162, -54, -18. And the average of it is -78 and it represents the average rate at which the number of contestants changed from the first round to the fourth round.