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The graph represents a periodic function, showing the population of deer as a function of time. The function appears to follow a sinusoidal pattern, which can be modeled using a sine or cosine function.

### Key observations:
1. **Amplitude**: The maximum population is approximately 800, and the minimum is about 200. So the amplitude is half the difference between these two values:
   \[
   A = \frac{800 - 200}{2} = 300
   \]

2. **Midline**: The midline (average value of the function) is the middle between 800 and 200:
   \[
   \text{Midline} = \frac{800 + 200}{2} = 500
   \]

3. **Period**: From the graph, the period (the time it takes to complete one full cycle) appears to be about 12 years (from peak to peak or trough to trough).

4. **Horizontal Shift**: The graph is a cosine-like curve but starts decreasing at \(x = 0\). Hence, a sine function might be more appropriate since sine typically starts at zero and increases or decreases, but it would need a phase shift.

### General equation for a sine function:
\[
f(x) = A \sin(B(x - C)) + D
\]
Where:
- \(A\) is the amplitude,
- \(B\) is related to the period by \(B = \frac{2\pi}{\text{Period}}\),
- \(C\) is the horizontal shift,
- \(D\) is the vertical shift (the midline).

### Applying the values:
- Amplitude \(A = 300\),
- Period = 12, so \(B = \frac{2\pi}{12} = \frac{\pi}{6}\),
- No clear horizontal shift for this model since the sine function fits naturally, so \(C = 0\),
- Midline \(D = 500\).

Thus, the rule of this function can be written as:
\[
f(x) = 300 \sin\left(\frac{\pi}{6}x\right) + 500
\]

This is the rule that represents the evolution of the deer population over time.

Would you like further details on how the period or amplitude was determined?

Here are 5 related questions to expand on this:
1. How would the function change if the population oscillated between 100 and 900 deer instead?
2. What would the function look like if the period was 8 years instead of 12?
3. How does the midline affect the overall behavior of the population graph?
4. What would the equation be if the graph started at its maximum instead of minimum?
5. Can we model this using a cosine function instead of sine?

**Tip**: Remember, in periodic functions like this, the period and amplitude play crucial roles in determining the shape and behavior of the graph.

Write the rule that can be used to represent this function. The graph shows a sinusoidal pattern representing the deer population as a function of time over 18 years.

Learn how to model the population of deer over time using a sinusoidal function. The graph shows a periodic pattern with key observations such as amplitude, midline, and period, forming the function f(x) = 300 * sin(π/6 * x) + 500.

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