To determine which type of function best models the relationship between the day and the number of visitors, we need to analyze the trends in the data.
Given data:
Days ([tex]\(x\)[/tex]):
[tex]\[1, 2, 3, 4, 5, 6, 7\][/tex]
Visitors ([tex]\(y\)[/tex]):
[tex]\[45, 86, 124, 138, 145, 158, 162\][/tex]
We want to find the best type of function to fit this data. Possible options include a linear function, a quadratic function (either with a positive or negative leading coefficient [tex]\(a\)[/tex]), and a square root function.
### Analysis:
1. Quadratic Function:
- A quadratic function typically has the form [tex]\(y = ax^2 + bx + c\)[/tex].
- The quadratic function can either open upwards (positive [tex]\(a\)[/tex]) or downwards (negative [tex]\(a\)[/tex]).
- Looking at the data, the increases in the number of visitors do not appear to be constant, suggesting a non-linear relationship, which makes quadratic functions worth considering.
2. Linear Function:
- A linear function has the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
- If the relationship between days and visitors was linear, the visitors would increase by a constant amount each day.
- Based on the provided data, the number of visitors does not increase by a constant amount each day, ruling out a simple linear function.
3. Square Root Function:
- A square root function has the form [tex]\(y = a\sqrt{x} + b\)[/tex].
- This type of function usually increases quickly at first and then levels off.
- The provided data does not fit this pattern well.
Given these points, the most plausible type of function that models the relationship between the day and the number of visitors appears to be quadratic.
Now, to decide between a positive or negative value of [tex]\(a\)[/tex]:
- If the quadratic function has a positive leading coefficient ([tex]\(a > 0\)[/tex]), it means the function opens upwards.
- If the quadratic function has a negative leading coefficient ([tex]\(a < 0\)[/tex]), it means the function opens downwards.
By examining the data, we observe a trend where the number of visitors increases more rapidly initially and then starts to level off. This behavior aligns with the nature of a quadratic function with a negative leading coefficient [tex]\(a\)[/tex].
Therefore, the best type of function to model this data is:
A. a quadratic function with a negative value of [tex]\(a\)[/tex].
