https://file.aiquickdraw.com/mathbot/file/1726521992368wqdvae3v.jpg

The graph in the image represents a sinusoidal function that models the daylight in a certain city over the course of 24 months. Let's analyze the graph and extract key information:

### Key points from the graph:
- **Peaks (Maxima):**
  - At \( t = 11.5 \) months, the daylight is at a maximum of 835 minutes.
  - At \( t = 23.5 \) months, the daylight again reaches 835 minutes.
  
- **Troughs (Minima):**
  - At \( t = 5.5 \) months, the daylight is at a minimum of 497 minutes.
  - At \( t = 17.5 \) months, the daylight is again at 497 minutes.

### Interpretation:
- The graph shows a **periodic pattern**, indicating that the function repeats every 12 months (since the peaks and troughs are separated by 12 months).
- This suggests a **cosine or sine function** can be used to model the average daylight over time.

### General form of the sinusoidal function:
The sinusoidal function can be written as:
\[
y(t) = A \cos(B(t - D)) + C
\]
Where:
- \( A \) is the amplitude (half the distance between the maximum and minimum values),
- \( B \) is related to the period of the function (specifically, \( B = \frac{2\pi}{\text{Period}} \)),
- \( D \) is the horizontal shift (phase shift),
- \( C \) is the vertical shift (midline of the function).

### Determining the constants:
1. **Amplitude (A):**
   \[
   A = \frac{835 - 497}{2} = 169 \, \text{minutes}
   \]

2. **Period:**
   The period is 12 months (since the pattern repeats every 12 months).
   \[
   B = \frac{2\pi}{12} = \frac{\pi}{6}
   \]

3. **Vertical shift (C):**
   The midline (average of the max and min values) is:
   \[
   C = \frac{835 + 497}{2} = 666 \, \text{minutes}
   \]

4. **Horizontal shift (D):**
   Since the maximum occurs at \( t = 11.5 \), the phase shift is 11.5 months. In the cosine function, we subtract this shift.

### Equation of the function:
Thus, the function modeling the average daylight \( y(t) \) in minutes, where \( t \) is in months, is:
\[
y(t) = 169 \cos\left(\frac{\pi}{6}(t - 11.5)\right) + 666
\]

Let me know if you'd like more details or have any questions.

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Here are 5 related questions:
1. How can we use this equation to predict the average daylight in a specific month?
2. How does the amplitude of this function relate to the variation in daylight across seasons?
3. Can this sinusoidal model be used for other cities with different daylight patterns?
4. What would happen to the equation if the city's daylight hours shifted due to climate changes?
5. How can we determine the months of equinoxes using this model?

**Tip:** Sinusoidal functions are great tools for modeling periodic phenomena like daylight, temperature, or tidal patterns because of their natural repeating structure!

Over the course of a full year, the daylight in a certain city follows a periodic pattern. The graph represents the average daylight, in minutes, over the course of 24 months, with time t representing the number of months after January 1 of a certain year.

Learn how to model the average daylight in a city over a 24-month period using sinusoidal functions. This problem involves concepts such as amplitude, phase shift, and vertical shift to predict periodic changes in daylight. Suitable for students in Grades 11-12.

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