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To find the function that matches the given graph, we need to analyze its amplitude, period, and vertical shift. Let's go through this step by step:

### Step 1: Identify the form of the equation
The form of the function is given as:
\[ y = A \sin(kx) + C \quad \text{or} \quad y = A \cos(kx) + C \]
Where:
- \( A \) is the amplitude.
- \( k \) affects the period (how stretched or compressed the wave is).
- \( C \) is the vertical shift.

### Step 2: Determine the amplitude \( A \)
From the graph, we observe that the maximum value is 0 and the minimum value is -4. The amplitude is half the distance between the maximum and minimum values:
\[
A = \frac{0 - (-4)}{2} = 2
\]
So, \( A = 2 \).

### Step 3: Determine the period and \( k \)
The period is the length of one complete cycle. From the graph, we see that one cycle occurs between \( x = -6 \) and \( x = 6 \), which gives a period of:
\[
\text{Period} = 12
\]
The period is related to \( k \) by the formula:
\[
\text{Period} = \frac{2\pi}{k}
\]
Thus, we can solve for \( k \):
\[
12 = \frac{2\pi}{k} \quad \Rightarrow \quad k = \frac{2\pi}{12} = \frac{\pi}{6}
\]

### Step 4: Determine the vertical shift \( C \)
The midline of the wave (the horizontal axis around which the sine or cosine oscillates) is at \( y = -2 \), so \( C = -2 \).

### Step 5: Choose the sine or cosine function
From the graph, it looks like the curve starts at the minimum point when \( x = 0 \). This behavior corresponds to a cosine function with a phase shift or a sine function that has been reflected. Since the graph looks like a downward-shifted cosine, we use the cosine form.

### Final equation
The function that matches the graph is:
\[
y = -2 \cos\left(\frac{\pi}{6}x\right) - 2
\]

Would you like a deeper explanation or have further questions on this? Here are 5 related questions to explore further:

1. How do we handle phase shifts in sinusoidal functions?
2. Can we also express the same function using a sine form?
3. What happens to the graph if we change the amplitude \( A \)?
4. How do we calculate the frequency from the period of the wave?
5. What is the significance of the vertical shift \( C \) in real-world applications?

**Tip:** For sinusoidal functions, the period is always given by \( \frac{2\pi}{k} \), so larger values of \( k \) lead to more compressed waves, while smaller values of \( k \) make the wave stretch out.

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

Discover how to find a trigonometric function in the form y = A*sin(kx) + C or y = A*cos(kx) + C by analyzing the amplitude, period, and vertical shift. Learn to match a function to a graph with a detailed explanation, suitable for Grades 10-12 students.

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