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.
To determine which function is the same type as W(x), we need to analyze the pattern in the given points.
- B(x) represents a linear function. Let's check if the points in W(x) form a straight line. We can calculate the slope between each set of consecutive points:
Slope between (3, 24) and (4, 48):
m1 = (48 - 24) / (4 - 3) = 24/1 = 24
Slope between (4, 48) and (5, 96):
m2 = (96 - 48) / (5 - 4) = 48/1 = 48
Slope between (5, 96) and (7, 384):
m3 = (384 - 96) / (7 - 5) = 288/2 = 144
Slope between (7, 384) and (8, 768):
m4 = (768 - 384) / (8 - 7) = 384/1 = 384
Slope between (8, 768) and (9, 1536):
m5 = (1536 - 768) / (9 - 8) = 768/1 = 768
Since the slopes between consecutive points in W(x) are not constant, W(x) is not a linear function. B(x) cannot be the same type as W(x).
- M(x) represents an exponential function of the form M(x) = a * r^x, where a is the initial value and r is the common ratio.
Let's check if the given points in W(x) follow an exponential pattern:
(3, 24): M(3) = a * r^3 = 24
(4, 48): M(4) = a * r^4 = 48
(5, 96): M(5) = a * r^5 = 96
(7, 384): M(7) = a * r^7 = 384
(8, 768): M(8) = a * r^8 = 768
(9, 1536): M(9) = a * r^9 = 1536
Dividing each equation by the previous one, we get the common ratio:
48/24 = r
96/48 = r
384/96 = r
768/384 = r
1536/768 = r
r = 2 in all cases, indicating that the points in W(x) follow an exponential pattern. Therefore, M(x) is the same type as W(x).
- S(x) represents a quadratic function. let's check if the given points in W(x) follow a quadratic pattern.
To do this, we need to look at the differences between consecutive outputs:
Difference between 24 and 48: 48 - 24 = 24
Difference between 48 and 96: 96 - 48 = 48
Difference between 96 and 384: 384 - 96 = 288
Difference between 384 and 768: 768 - 384 = 384
Difference between 768 and 1536: 1536 - 768 = 768
The differences are not constant, indicating that the points in W(x) do not follow a quadratic pattern. Therefore, S(x) is not the same type as W(x).
In conclusion, the function M(x) is the same type as W(x) because both follow an exponential pattern, where the common ratio is 2.
To determine which function (B(x), M(x), or S(x)) is the same type as W(x), we need to analyze the pattern in the given points for W(x) and compare it with the patterns of the other functions.
Looking at the points for W(x):
(3, 24), (4, 48), (5, 96), (7, 384), (8, 768), and (9, 1536)
We can observe that in each consecutive point, the y-value doubles. For example:
24 * 2 = 48
48 * 2 = 96
96 * 2 = 192 (missing point 6, which we can notice from the pattern)
192 * 2 = 384
384 * 2 = 768
768 * 2 = 1536
This indicates an exponential growth pattern with a base of 2.
Now let's analyze the patterns for B(x), M(x), and S(x) to determine which one matches the pattern of W(x):
B(x): This represents a linear growth pattern, where each consecutive y-value increases by the same amount. For example, if we subtract points (3, 2) and (2, 1), we get:
2 - 1 = 1
This means that for B(x), the y-value increases by 1 for each unit increase in x. This pattern does not match W(x) because in W(x), the y-value doubles with each increase in x.
M(x): This represents an exponential growth pattern, where each consecutive y-value is multiplied by the same factor. For example, if we divide points (4, 2500) and (3, 500), we get:
2500 / 500 = 5
This means that for M(x), the y-value is multiplied by 5 for each unit increase in x. This pattern does not match W(x) because in W(x), the y-value doubles with each increase in x, not multiplied by a fixed factor.
S(x): This represents an exponential growth pattern, where each consecutive y-value is multiplied by the same factor. For example, if we divide points (2, 5) and (1, 2), we get:
5 / 2 = 2.5
This means that for S(x), the y-value is multiplied by 2.5 for each unit increase in x. This pattern does not match W(x) because in W(x), the y-value doubles with each increase in x, not multiplied by a fixed factor.
From our analysis, we can conclude that the function W(x) has an exponential growth pattern with a base of 2, matching the pattern of function B(x). Therefore, B(x) is the same type as W(x).