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To find a trigonometric function that matches the graph in the form \( y = A \sin(kx) + C \) or \( y = A \cos(kx) + C \), we can break down the components based on the properties of the graph.

### Key properties to observe:
1. **Amplitude (A):** The graph oscillates between 5 and -5, which means the amplitude is 5. So, \( A = 5 \).
   
2. **Vertical Shift (C):** The midline of the graph is at \( y = 0 \), so there is no vertical shift. Thus, \( C = 0 \).

3. **Period (T):** To find the period, observe the x-axis. One full cycle of the graph (from peak to peak) spans from \( x = -12 \) to \( x = 0 \), so the period \( T \) is 12. The relationship between period \( T \) and \( k \) is given by the formula \( T = \frac{2\pi}{k} \). Solving for \( k \), we get:
   \[
   T = 12 \implies \frac{2\pi}{k} = 12 \implies k = \frac{2\pi}{12} = \frac{\pi}{6}.
   \]

4. **Phase Shift:** Since the graph looks like a cosine function starting at a peak at \( x = -12 \), we will use the cosine form of the function \( y = A \cos(kx) + C \) with no phase shift required.

### The resulting function:
\[
y = 5 \cos\left(\frac{\pi}{6} x\right)
\]

Would you like any further clarification or details on this process? 

Here are five related questions:
1. How would the function change if there were a phase shift?
2. Can we model this graph using a sine function instead of a cosine function?
3. How would the function change if the amplitude were doubled?
4. What would happen to the period if \( k \) were doubled?
5. How do vertical shifts affect the graph of trigonometric functions?

**Tip:** When analyzing trigonometric graphs, always check the amplitude, period, and any shifts to determine the equation correctly.

Find a function of the form y = A*sin(kx) + C or y = A*cos(kx) + C whose graph matches the function shown below.

Learn how to match a trigonometric graph to a function of the form y = A*sin(kx) + C or y = A*cos(kx) + C. This problem covers key concepts in trigonometry, including amplitude, period, and how to determine the equation of a sinusoidal graph. Suitable for high school students in Grades 10-12.

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