Answer :
To determine which type of function best models the relationship between the day and the number of visitors, we need to consider how the number of visitors changes over the days.
Given the data:
[tex]\[ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text{Day} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Visitors} & 45 & 86 & 124 & 138 & 145 & 158 & 162 \\ \hline \end{array} \][/tex]
### Step-by-step Analysis:
1. List Differences Between Consecutive Visitor Counts:
- Day 2 - Day 1: [tex]\(86 - 45 = 41\)[/tex]
- Day 3 - Day 2: [tex]\(124 - 86 = 38\)[/tex]
- Day 4 - Day 3: [tex]\(138 - 124 = 14\)[/tex]
- Day 5 - Day 4: [tex]\(145 - 138 = 7\)[/tex]
- Day 6 - Day 5: [tex]\(158 - 145 = 13\)[/tex]
- Day 7 - Day 6: [tex]\(162 - 158 = 4\)[/tex]
2. Check for Linearity:
- For a linear function, the differences between consecutive visitor counts should be approximately constant. Here, they are not consistent.
3. Consider a Quadratic Function:
- Differences between differences:
- [tex]\( 38 - 41 = -3 \)[/tex]
- [tex]\( 14 - 38 = -24\)[/tex]
- [tex]\( 7 - 14 = -7\)[/tex]
- [tex]\( 13 - 7 = 6\)[/tex]
- [tex]\( 4 - 13 = -9\)[/tex]
- The second set of differences (second derivative) is not constant or nearly constant which would be expected in a well-fitting quadratic function.
### Type of Functions Considered:
- Linear Function with a Positive Slope (Option A):
- Not appropriate as the differences are not consistent.
- Quadratic Function with a Positive Value of [tex]\(a\)[/tex] (Option B):
- This suggests an upward curvature (increasing rate of increase). The differences are not constant but the growth slows down and picks up irregularly which fits a quadratic model better.
- Quadratic Function with a Negative Value of [tex]\(a\)[/tex] (Option C):
- This would imply a parabolic shape opening downward (decreasing rate of increase after increasing). This is not observed in the data.
- Square Root Function (Option D):
- Typically shows rapid initial growth then levels out; our data shows a more complex pattern.
Based on the observation and consideration:
### Conclusion:
The best fitting function is most likely:
Option B: a quadratic function with a positive value of [tex]\(a\)[/tex].
This indicates the number of visitors initially increases rapidly (possibility an effect of quadratic behavior), slows, and then picks up again, aligning more with quadratic characteristics rather than linear or square root patterns.
Given the data:
[tex]\[ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text{Day} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Visitors} & 45 & 86 & 124 & 138 & 145 & 158 & 162 \\ \hline \end{array} \][/tex]
### Step-by-step Analysis:
1. List Differences Between Consecutive Visitor Counts:
- Day 2 - Day 1: [tex]\(86 - 45 = 41\)[/tex]
- Day 3 - Day 2: [tex]\(124 - 86 = 38\)[/tex]
- Day 4 - Day 3: [tex]\(138 - 124 = 14\)[/tex]
- Day 5 - Day 4: [tex]\(145 - 138 = 7\)[/tex]
- Day 6 - Day 5: [tex]\(158 - 145 = 13\)[/tex]
- Day 7 - Day 6: [tex]\(162 - 158 = 4\)[/tex]
2. Check for Linearity:
- For a linear function, the differences between consecutive visitor counts should be approximately constant. Here, they are not consistent.
3. Consider a Quadratic Function:
- Differences between differences:
- [tex]\( 38 - 41 = -3 \)[/tex]
- [tex]\( 14 - 38 = -24\)[/tex]
- [tex]\( 7 - 14 = -7\)[/tex]
- [tex]\( 13 - 7 = 6\)[/tex]
- [tex]\( 4 - 13 = -9\)[/tex]
- The second set of differences (second derivative) is not constant or nearly constant which would be expected in a well-fitting quadratic function.
### Type of Functions Considered:
- Linear Function with a Positive Slope (Option A):
- Not appropriate as the differences are not consistent.
- Quadratic Function with a Positive Value of [tex]\(a\)[/tex] (Option B):
- This suggests an upward curvature (increasing rate of increase). The differences are not constant but the growth slows down and picks up irregularly which fits a quadratic model better.
- Quadratic Function with a Negative Value of [tex]\(a\)[/tex] (Option C):
- This would imply a parabolic shape opening downward (decreasing rate of increase after increasing). This is not observed in the data.
- Square Root Function (Option D):
- Typically shows rapid initial growth then levels out; our data shows a more complex pattern.
Based on the observation and consideration:
### Conclusion:
The best fitting function is most likely:
Option B: a quadratic function with a positive value of [tex]\(a\)[/tex].
This indicates the number of visitors initially increases rapidly (possibility an effect of quadratic behavior), slows, and then picks up again, aligning more with quadratic characteristics rather than linear or square root patterns.
