Find answers to your questions faster and easier with IDNLearn.com. Join our knowledgeable community to find the answers you need for any topic or issue.
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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
