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We are asked to find a trigonometric function of the form \( y = A \sin(kx) + C \) or \( y = A \cos(kx) + C \) that matches the graph shown.

### Steps to determine the equation:

1. **Amplitude \(A\)**: 
   The amplitude is the distance from the centerline (vertical shift) to the peak. From the graph, the maximum value is \( 8 \) and the minimum value is \( -8 \), so the amplitude is:
   \[
   A = \frac{8 - (-8)}{2} = 8
   \]

2. **Period and \( k \)**: 
   The period is the distance between two consecutive peaks or troughs. Observing the graph, the peaks are at \( x = -9 \) and \( x = -3 \), meaning the period is:
   \[
   \text{Period} = -3 - (-9) = 6
   \]
   The relationship between the period and \( k \) is given by:
   \[
   \text{Period} = \frac{2\pi}{k}
   \]
   So, solving for \( k \):
   \[
   6 = \frac{2\pi}{k} \quad \Rightarrow \quad k = \frac{2\pi}{6} = \frac{\pi}{3}
   \]

3. **Phase Shift**:
   The graph resembles a cosine wave, as it starts at a maximum at \( x = -9 \). Since the general cosine function \( \cos(kx) \) has its peak at \( x = 0 \), this graph is shifted to the left by 9 units. Therefore, the phase shift is \( 9 \).

4. **Vertical Shift \( C \)**:
   The graph oscillates symmetrically about \( y = 0 \), so there is no vertical shift. Thus, \( C = 0 \).

### Final Equation:
The equation that matches the graph is:
\[
y = 8 \cos\left( \frac{\pi}{3}(x + 9) \right)
\]

Let me know if you need further details or explanations!

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Here are 5 related questions you can explore:
1. How do we calculate the amplitude for a trigonometric function?
2. What is the general formula for the period of sine and cosine functions?
3. How does the phase shift affect the graph of a trigonometric function?
4. Can this graph be represented using a sine function instead of a cosine function?
5. What role does the vertical shift \(C\) play in modifying a trigonometric function?

**Tip**: The cosine function peaks at \( x = 0 \), while the sine function starts from 0, which helps decide which form to use when modeling real-life periodic graphs.

Find a function of the form y = A sin(kx) + C or y = A cos(kx) + C whose graph matches the function shown below. Leave your answer in exact form; if necessary, type pi for π.

Learn how to find a trigonometric function that matches a given graph. This problem involves determining amplitude, period, phase shift, and vertical shifts for functions of the form y = A sin(kx) + C or y = A cos(kx) + C. Ideal for students in Grades 9-12.

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