Question

select the correct answer from each drop - down menu. the number of people who visited a state park over the last nine years is recorded in the table. year 1 2 3 4 5 6 7 8 9 visitors 452 360 348 302 198 124 204 308 398 complete the statements describing the situation. the visitor data is best modeled by a function. based on the model, the state park management should plan for than 400 visitors next year

Explanation:

Step 1: Analyze the visitor data trend

First, we observe the number of visitors over the years: 452, 360, 348, 302, 198, 124, 204, 308, 398. The data decreases to a minimum (124 at year 6) and then increases again. This pattern is characteristic of a quadratic (or parabolic) function, which has a minimum (or maximum) point and then changes direction. Linear functions have a constant rate of change (either always increasing or always decreasing), and exponential functions have a multiplicative rate of change (either increasing or decreasing rapidly). Since the data decreases then increases, a quadratic function is the best model.

Step 2: Predict the next year's visitors

To predict the next year (year 10), we can consider the symmetry of the quadratic function. The minimum is at year 6 (124 visitors). The distance from year 6 to year 1 is \(6 - 1 = 5\) years, and the distance from year 6 to year 10 is also \(10 - 6 = 4\) years (close to 5, showing symmetry). Let's check the symmetry around year 6:

• Year 5 (198) and year 7 (204) are symmetric around year 6: \(6 - 5 = 1\), \(7 - 6 = 1\). The values are 198 and 204 (close).
• Year 4 (302) and year 8 (308): \(6 - 4 = 2\), \(8 - 6 = 2\). Values 302 and 308 (close).
• Year 3 (348) and year 9 (398): \(6 - 3 = 3\), \(9 - 6 = 3\). Wait, year 3 is 348, year 9 is 398. Wait, maybe better to check the increase after year 6. From year 6 (124) to year 7 (204): increase by 80. Year 7 to 8: increase by 104. Year 8 to 9: increase by 90. From year 1 to 6: decrease by 452 - 124 = 328 over 5 years. From year 6 to 9: increase by 398 - 124 = 274 over 3 years. If we assume the quadratic model, the value at year 10 should be compared to year 0 (if we extend), but since year 1 is 452, year 9 is 398. The increase from year 9 (398) to year 10: following the pattern of the quadratic (increasing after the minimum), but let's see the symmetry around the vertex (year 6). The value at year 1 (452) and year 11 (if we go 5 years after year 6) would be symmetric, but year 10 is 4 years after year 6. Alternatively, we can see that the increase from year 8 (308) to year 9 (398) is 90, so if we assume a similar or slightly decreasing increase (since the quadratic's rate of change changes), the next year (year 10) would be around 398 + (90 - some amount) or 398 + similar. But since the maximum before the decrease was year 1 (452), and now at year 9 it's 398 (less than 452), the increase after year 6 is not reaching back to 452 yet. So the number of visitors at year 10 should be more than 398 but let's check the symmetry. The distance from year 6 to year 1 is 5, year 6 to year 10 is 4. The value at year 1 is 452, so the value at year 10 should be less than 452, but let's see the trend after year 6: year 7 (204), year 8 (308), year 9 (398). The differences between consecutive years after year 6: 204 - 124 = 80, 308 - 204 = 104, 398 - 308 = 90. So the increases are 80, 104, 90. If we assume the next increase is around 80 - 10 (since it's a quadratic, the second difference should be constant). The second differences for the decreasing part: 452 to 360: -92; 360 to 348: -12; 348 to 302: -46; 302 to 198: -104; 198 to 124: -74; 124 to 204: +80; 204 to 308: +104; 308 to 398: +90. The second differences (difference of differences) for the decreasing part: -12 - (-92) = 80; -46 - (-12) = -34; -104 - (-46) = -58; -74 - (-104) = 30. For the increasing part: 104 - 80 = 24; 90 - 104 = -14. This is a bit messy, but the key is the data has a minimum, so quadratic. Now, to predict year 10: since year 9 is 398, and the trend after…

Step 1: Identify the best model

The visitor data decreases to a minimum (year 6, 124) then increases, indicating a quadratic (parabolic) trend (linear is constant slope, exponential is multiplicative, but this has a minimum and changes direction).

Step 2: Predict next year’s visitors

Using the quadratic symmetry around year 6 (vertex), the value at year 10 (symmetric to year 2 with a slight adjustment) is less than 400 (e.g., via quadratic model calculation or trend analysis).

Answer:

The visitor data is best modeled by a \(\boldsymbol{\text{quadratic}}\) function.
Based on the model, the state park management should plan for \(\boldsymbol{\text{less}}\) than 400 visitors next year.