To determine which type of function best models the data, let's investigate the relationship between the years after opening and the number of employees by fitting three different types of models – linear, quadratic (both with positive and negative [tex]\(a\)[/tex]), and square root models. Here is a step-by-step approach:
1. Data Representation:
- Years After Opening: [tex]\([1, 2, 3, 4, 5, 6, 7]\)[/tex]
- Number of Employees: [tex]\([100, 348, 405, 575, 654, 704, 746]\)[/tex]
2. Model Fitting:
- Linear Model: This assumes a relationship of the form [tex]\( y = mx + c \)[/tex].
- Quadratic Model: This assumes a relationship of the form [tex]\( y = ax^2 + bx + c \)[/tex].
- Square Root Model: This assumes a relationship of the form [tex]\( y = a \sqrt{x} + b \)[/tex].
3. Residuals Calculation:
Residuals are the differences between the actual number of employees and the predicted number according to each model. The sum of the squared residuals is a measure of how well the model fits the data; the smaller the sum, the better the model fits.
4. Results:
- Sum of squared residuals for the Linear Model: 23,305.678571428587
- Sum of squared residuals for the Quadratic Model: 4,793.66666666667
- Sum of squared residuals for the Square Root Model: 8,407.09737584916
5. Best Fit Selection:
- Among these models, the Quadratic Model has the smallest sum of squared residuals (4,793.66666666667), indicating it provides the best fit for the data.
6. Determine Quadratic Coefficient [tex]\(a\)[/tex]:
- We now determine the nature of the quadratic function. Given that the quadratic model offers the best fit, we examine the coefficient [tex]\(a\)[/tex] of the quadratic term. Since it provided the best fit and is associated with a positive value of [tex]\(a\)[/tex], it means the model is best described by a quadratic function with a negative value of [tex]\(a\)[/tex].
Therefore, the correct answer is:
C. a quadratic function with a negative value of a
