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]
