https://file.aiquickdraw.com/mathbot/file/17242998327939pxau2vy.jpg

Let's analyze and complete the information for graphs 11 and 12 as requested.

### **Graph 11:**

1. **Amplitude \( a \):**  
   The amplitude is the distance from the midline of the graph to a peak (maximum or minimum).  
   **Amplitude \( a = 3 \)**

2. **Period \( d \):**  
   The period is the distance required for the function to complete one full cycle.  
   **Period \( d = 6 \)**

3. **Phase shift for cosine:**  
   Identify where the cosine function starts relative to its normal position at \( x = 0 \). Here, the cosine graph is shifted **right** by \( \pi/3 \).  
   **Phase shift for cosine = Right \( \frac{\pi}{3} \)**

4. **Phase shift for sine:**  
   The sine function is shifted **right** by \( \pi/6 \).  
   **Phase shift for sine = Right \( \frac{\pi}{6} \)**

5. **Function equation:**  
   Using sine:  
   \( f(x) = 3 \sin \left(\frac{\pi}{3}x - \frac{\pi}{6}\right) \)  
   Using cosine:  
   \( f(x) = 3 \cos \left(\frac{\pi}{3}x - \frac{\pi}{3}\right) \)

### **Graph 12:**

1. **Amplitude \( a \):**  
   **Amplitude \( a = 2 \)**

2. **Period \( d \):**  
   **Period \( d = 8 \)**

3. **Phase shift for cosine:**  
   The graph shifts **left** by \( 2 \).  
   **Phase shift for cosine = Left \( 2 \)**

4. **Phase shift for sine:**  
   The graph shifts **right** by \( 4 \).  
   **Phase shift for sine = Right \( 4 \)**

5. **Function equation:**  
   Using sine:  
   \( f(x) = 2 \sin \left(\frac{\pi}{4}x - 4\right) \)  
   Using cosine:  
   \( f(x) = 2 \cos \left(\frac{\pi}{4}x + 2\right) \)

### Summary:
- **Graph 11:** Amplitude \( 3 \), Period \( 6 \), Phase shifts: \( \frac{\pi}{3} \) for cosine, \( \frac{\pi}{6} \) for sine.
- **Graph 12:** Amplitude \( 2 \), Period \( 8 \), Phase shifts: Left 2 for cosine, Right 4 for sine.

Would you like more details on these calculations, or do you have any other questions?

### Follow-Up Questions:
1. How do you determine which function (sine or cosine) is more appropriate to use based on the graph?
2. What is the effect of changing the amplitude on the graph?
3. How can you use the graph to determine vertical shifts?
4. How does the phase shift affect the starting point of the function?
5. What role does the period play in determining the frequency of a trigonometric graph?

### Tip:
Always align the graph's key points (like maximum, minimum, and intercepts) with the corresponding standard sine or cosine curve to identify the phase shift accurately.

Explore detailed analysis and solutions for trigonometric graphs, focusing on amplitude, period, and phase shifts. Includes examples of cosine and sine functions with specific calculations. Ideal for high school students learning trigonometry.

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