To determine which type of function best models the relationship between the day and the number of visitors, we need to examine the trend shown by the numbers of visitors over the given days.
Let's look at the number of visitors over the course of the week:
- Day 1: 45
- Day 2: 86
- Day 3: 124
- Day 4: 138
- Day 5: 145
- Day 6: 158
- Day 7: 162
Next, let's observe whether the increases in the number of visitors indicate a specific type of function:
1. Linear function with a positive slope:
A linear function would have a constant rate of change. We can check the differences between the consecutive days:
- Day 2 - Day 1 = 86 - 45 = 41
- Day 3 - Day 2 = 124 - 86 = 38
- Day 4 - Day 3 = 138 - 124 = 14
- Day 5 - Day 4 = 145 - 138 = 7
- Day 6 - Day 5 = 158 - 145 = 13
- Day 7 - Day 6 = 162 - 158 = 4
Since the differences aren't constant, it's unlikely to be a linear function.
2. Quadratic function with a positive value of \(a\):
A quadratic function with a positive value of \(a\) would generally show an increasing pattern at an increasing rate. Let's reflect on the trend:
- The first increase (41, 38) decreases somewhat.
- However, later on, the increases significantly drop (14, 7).
While the overall trend is increasing, it doesn't strongly suggest the rapid acceleration characteristic of positive quadratic functions.
3. Quadratic function with a negative value of \(a\):
A quadratic function with a negative \(a\) value would show an initial increase then decrease, forming an inverted U-shape pattern. This pattern isn’t supported by the continuously increasing but incrementally smaller values observed.
4. Square root function:
A square root function might initially show larger increases that taper off over time. This pattern seems more fitting, as the increases diminish:
- Early increases in visitors are large (41, 38), followed by significantly smaller increments (14, 7).
In summary, a square root function is most consistent with the observed decreases in successive increases in visitors, as it initially rises quickly and then levels off.
Based on the detailed examination, the type of function that best models the relationship is:
C. a square root function
