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.
To determine the type of function, we need to analyze the relationship between the years (x) and the corresponding values of B(x), M(x), and S(x).
For B(x):
From the given data, we can see that as the years (x) increase, the values of B(x) also increase but not exponentially. The values seem to be increasing linearly, as the values of B(x) are being multiplied by a constant factor of 3 each year (2 * 3 = 6, 6 * 3 = 18, 18 * 3 = 54, and so on). Therefore, B(x) is a linear function.
For M(x):
From the given data, we can see that as the years (x) increase, the values of M(x) also increase but not linearly. The values seem to be increasing at an increasing rate, suggesting a quadratic relationship. To confirm this, we can calculate the ratios of consecutive terms: (120-20)/(6-2) = 100/4 = 25 and (420-120)/(18-6) = 300/12 = 25. Since the ratios are constant, it indicates a quadratic relationship, and therefore, M(x) is a quadratic function.
For S(x):
From the given data, we can see that as the years (x) increase, the values of S(x) increase rapidly. The values seem to be increasing exponentially, as each term is close to being tripled from the previous term (38 * 3 = 114, 81 * 3 = 243, 124 * 3 = 372, and so on). To confirm this, we can calculate the ratios of consecutive terms: 81/38 ≈ 2.13, 124/81 ≈ 1.53, 167/124 ≈ 1.35. Since the ratios are not constant, it indicates an exponential relationship, and therefore, S(x) is an exponential function.
Regarding the white-handed gibbon population:
The given data does not provide any information about the white-handed gibbon population (W(x)). Therefore, we cannot determine its type of function without further information or data.
To determine the type of function for B(x), M(x), and S(x), we can analyze the given data and look for patterns or relationships between the years (x) and the corresponding values.
1. B(x) - Batana:
From the given data, we can observe the following relationship:
B(0) = 2
B(1) = 6
B(2) = 18
B(3) = 54
B(4) = 162
B(5) = 486
Next, let's calculate the ratios between consecutive terms:
B(1)/B(0) = 6/2 = 3
B(2)/B(1) = 18/6 = 3
B(3)/B(2) = 54/18 = 3
B(4)/B(3) = 162/54 = 3
B(5)/B(4) = 486/162 = 3
The ratios between consecutive terms are all equal to 3. This indicates an exponential relationship since the values are increasing by a constant ratio.
Therefore, B(x) is an exponential function.
2. M(x) - Misool:
Using similar calculations, we can analyze the given data for M(x):
M(0) = 20
M(1) = 120
M(2) = 420
M(3) = 920
M(4) = 1620
M(5) = 2520
Calculating the ratios:
M(1)/M(0) = 120/20 = 6
M(2)/M(1) = 420/120 = 3.5
M(3)/M(2) = 920/420 ≈ 2.19
M(4)/M(3) = 1620/920 ≈ 1.76
M(5)/M(4) = 2520/1620 ≈ 1.56
The ratios between consecutive terms are not constant. A quadratic function is a polynomial of degree 2, and looking at the data, the ratios seem to decrease from term to term, which suggests a quadratic relationship.
Therefore, M(x) is a quadratic function.
3. S(x) - Salawati:
Analyzing the data for S(x):
S(0) = 38
S(1) = 81
S(2) = 124
S(3) = 167
S(4) = 210
S(5) = 253
Calculating the differences between consecutive terms:
81 - 38 = 43
124 - 81 = 43
167 - 124 = 43
210 - 167 = 43
253 - 210 = 43
The differences between consecutive terms are all equal to 43. This indicates a linear relationship since the values are increasing by a constant difference.
Therefore, S(x) is a linear function.
4. W(x) - White-handed Gibbon Population:
The population of white-handed gibbons, represented by W(x), at the fourth location starting in year three is not given in the provided data. Therefore, we cannot determine the function type for W(x) without additional information.