Question

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Scientists are studying the population of a rare species of primates, known as a Javan gibbons, on the Raja Ampat Islands in Indonesia. They are studying the gibbons on three different islands, Batanta, Misool, and Salawati. The scientists studied the population growth and recorded the data in the table.

Years (x) Batana
B(x) Misool
M(x) Salawati
S(x)
0 2 20 38
1 6 120 81
2 18 420 124
3 54 920 167
4 162 1620 210
5 486 2520 253


What type of function is B(x), linear, quadratic or exponential? Justify your answer and show calculations to support your conclusion.

What type of function is M(x)? Justify your answer and show calculations to support your conclusion.

What type of function is S(x)? Justify your answer and show calculations to support your conclusion.

A white-handed gibbon population, represented by W(x), was recorded at a fourth location, starting in year three.

The graph shows the function w(x), which includes five points. The points are located at (3, 24), (4, 48), (5, 96), (7, 384), (8, 768), and (9, 1536).

Of the functions B(x), M(x), and S(x), which function is the same type as W(x)? Justify your answer and show calculations to support your conclusion.

Answer 1

For Batanta island (B(x)):

The data for B(x) seems to increase linearly, as each value is three times the previous value (2, 6, 18, 54, 162, 486). We can check this by calculating the difference between consecutive values:

6 - 2 = 4

18 - 6 = 12

54 - 18 = 36

162 - 54 = 108

486 - 162 = 324

The differences between consecutive values increase by a factor of 3, which supports the idea that B(x) is a linear function.

For Misool island (M(x)):

The data for M(x) seems to increase exponentially, as each value is roughly twice the previous value (20, 120, 420, 920, 1620, 2520). We can check this by calculating the ratio between consecutive values:

120 / 20 = 6

420 / 120 = 3.5

920 / 420 = 2.19

1620 / 920 = 1.76

2520 / 1620 = 1.56

The ratios between consecutive values decrease as x increases, which supports the idea that M(x) is an exponential function.

For Salawati island (S(x)):

The data for S(x) seems to increase exponentially, as each value is roughly 1.5 times the previous value (38, 81, 124, 167, 210, 253). We can check this by calculating the ratio between consecutive values:

81 / 38 = 2.13

124 / 81 = 1.53

167 / 124 = 1.34

210 / 167 = 1.26

253 / 210 = 1.2

The ratios between consecutive values decrease as x increases, which supports the idea that S(x) is an exponential function.

For W(x):

Based on the given data points, it seems that W(x) is an exponential function. We can check this by calculating the ratio between consecutive values:

48 / 24 = 2

96 / 48 = 2

384 / 96 = 4

768 / 384 = 2

1536 / 768 = 2

The ratios between consecutive values remain constant at 2, which supports the idea that W(x) is an exponential function.

Therefore, of the functions B(x), M(x), and S(x), the same type of function as W(x) is M(x), since both are exponential functions.

Hope this helps!