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Sagot :
To determine the type of function that best models the data of the company's workforce growth, let's analyze the given information step by step.
1. Construct the Differences Between Consecutive Years:
Calculate the differences between the number of employees for consecutive years to understand the growth pattern.
[tex]\[ \begin{array}{c|c} \text{Years} & \text{Difference} \\ \hline 2-1 & 348 - 100 = 248 \\ 3-2 & 405 - 348 = 57 \\ 4-3 & 575 - 405 = 170 \\ 5-4 & 654 - 575 = 79 \\ 6-5 & 704 - 654 = 50 \\ 7-6 & 746 - 704 = 42 \\ \end{array} \][/tex]
So the differences between consecutive years are:
[tex]\[ [248, 57, 170, 79, 50, 42] \][/tex]
2. Construct the Second Differences (Differences of Differences):
Calculate the differences between consecutive differences to understand if the change is consistent (indicative of a quadratic function).
[tex]\[ \begin{array}{c|c} \text{Years} & \text{Second Difference} \\ \hline 3-2 & 57 - 248 = -191 \\ 4-3 & 170 - 57 = 113 \\ 5-4 & 79 - 170 = -91 \\ 6-5 & 50 - 79 = -29 \\ 7-6 & 42 - 50 = -8 \\ \end{array} \][/tex]
So the second differences are:
[tex]\[ [-191, 113, -91, -29, -8] \][/tex]
3. Evaluate Consistency of Second Differences:
For the data to be modeled well by a quadratic function, the second differences should be consistent.
Given:
[tex]\[ [-191, 113, -91, -29, -8] \][/tex]
The second differences are not consistent, suggesting that the function might not be a pure quadratic function straightforwardly.
4. Choosing the Best Function Model:
- A (Square Root Function): Typically involves a slower growth initially followed by faster growth, which doesn't fit our data.
- B (Quadratic Function with Negative [tex]\(a\)[/tex]): This would suggest the number of employees might eventually decrease, which isn't supported by our data.
- C (Linear Function with Positive Slope): Linear functions imply equal increments year over year, which isn’t the case here as the differences fluctuate considerably.
- D (Quadratic Function with Positive [tex]\(a\)[/tex]): Despite the second differences not being perfectly consistent, we observed that the rate of change is initially rapid and then slows down, a behavior characteristic of a quadratic function with a positive value of [tex]\(a\)[/tex].
Upon considering all these points, the type of function that best fits the data is:
[tex]\[ \boxed{D \text{. a quadratic function with a positive value of } a} \][/tex]
1. Construct the Differences Between Consecutive Years:
Calculate the differences between the number of employees for consecutive years to understand the growth pattern.
[tex]\[ \begin{array}{c|c} \text{Years} & \text{Difference} \\ \hline 2-1 & 348 - 100 = 248 \\ 3-2 & 405 - 348 = 57 \\ 4-3 & 575 - 405 = 170 \\ 5-4 & 654 - 575 = 79 \\ 6-5 & 704 - 654 = 50 \\ 7-6 & 746 - 704 = 42 \\ \end{array} \][/tex]
So the differences between consecutive years are:
[tex]\[ [248, 57, 170, 79, 50, 42] \][/tex]
2. Construct the Second Differences (Differences of Differences):
Calculate the differences between consecutive differences to understand if the change is consistent (indicative of a quadratic function).
[tex]\[ \begin{array}{c|c} \text{Years} & \text{Second Difference} \\ \hline 3-2 & 57 - 248 = -191 \\ 4-3 & 170 - 57 = 113 \\ 5-4 & 79 - 170 = -91 \\ 6-5 & 50 - 79 = -29 \\ 7-6 & 42 - 50 = -8 \\ \end{array} \][/tex]
So the second differences are:
[tex]\[ [-191, 113, -91, -29, -8] \][/tex]
3. Evaluate Consistency of Second Differences:
For the data to be modeled well by a quadratic function, the second differences should be consistent.
Given:
[tex]\[ [-191, 113, -91, -29, -8] \][/tex]
The second differences are not consistent, suggesting that the function might not be a pure quadratic function straightforwardly.
4. Choosing the Best Function Model:
- A (Square Root Function): Typically involves a slower growth initially followed by faster growth, which doesn't fit our data.
- B (Quadratic Function with Negative [tex]\(a\)[/tex]): This would suggest the number of employees might eventually decrease, which isn't supported by our data.
- C (Linear Function with Positive Slope): Linear functions imply equal increments year over year, which isn’t the case here as the differences fluctuate considerably.
- D (Quadratic Function with Positive [tex]\(a\)[/tex]): Despite the second differences not being perfectly consistent, we observed that the rate of change is initially rapid and then slows down, a behavior characteristic of a quadratic function with a positive value of [tex]\(a\)[/tex].
Upon considering all these points, the type of function that best fits the data is:
[tex]\[ \boxed{D \text{. a quadratic function with a positive value of } a} \][/tex]
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