Answer:
Explanation:
To determine the type of function M(x), we need to analyze the ratios, first differences, and second differences of the given data.
Ratios:
The ratios represent the rate of change between consecutive values of M(x). To find the ratios, divide each M(x) value by its corresponding x value.
Ratios:
M(1)/1 = 120/1 = 120
M(2)/2 = 420/2 = 210
M(3)/3 = 920/3 = 306.67
M(4)/4 = 1620/4 = 405
M(5)/5 = 2520/5 = 504
The ratios are not constant, which suggests that the function is not linear.
First Differences:
The first differences represent the change in M(x) between consecutive years (x values). To find the first differences, subtract each M(x) value from the previous M(x) value.
First Differences:
M(1) - M(0) = 120 - 20 = 100
M(2) - M(1) = 420 - 120 = 300
M(3) - M(2) = 920 - 420 = 500
M(4) - M(3) = 1620 - 920 = 700
M(5) - M(4) = 2520 - 1620 = 900
The first differences are not constant, which further confirms that the function is not linear.
Second Differences:
The second differences represent the change in the first differences between consecutive years (x values). To find the second differences, subtract each first difference value from the previous first difference value.
Second Differences:
(100 - 0) = 100
(300 - 100) = 200
(500 - 300) = 200
(700 - 500) = 200
(900 - 700) = 200
The second differences are constant. This suggests that the function may be a quadratic function.
Based on the analysis of the ratios, first differences, and second differences, we can conclude that the function M(x) is likely a quadratic function. A quadratic function has a second-degree polynomial expression, which can be written in the form M(x) = ax^2 + bx + c.
To find the specific quadratic equation for M(x), we can use the second differences to determine the coefficient of the quadratic term. Since the second differences are constant at 200, it means the coefficient of the quadratic term (a) is 200/2 = 100.
Therefore, the quadratic function M(x) is given by:
M(x) = 100x^2 + bx + c
Now, to find the values of b and c, we can use the data provided:
Using the point (0, 20):
M(0) = 100(0)^2 + b(0) + c
20 = 0 + 0 + c
c = 20
Using the point (1, 120):
M(1) = 100(1)^2 + b(1) + 20
120 = 100 + b + 20
b = 120 - 100 - 20
b = 0
Therefore, the quadratic function M(x) is:
M(x) = 100x^2 + 20
This confirms that the function M(x) is indeed a quadratic function.
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