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To determine the formula for the given cosine function, we need to focus on several aspects of the graph, such as amplitude, period, phase shift, and vertical shift.

### Step 1: **Amplitude**
The amplitude is the distance from the midline of the graph to the peak. From the graph, the peak is at 3.4 and the minimum is at 0.7. The midline, or average of these, is:
\[
\text{Midline} = \frac{3.4 + 0.7}{2} = 2.05
\]
The amplitude is the distance from this midline to the maximum or minimum value:
\[
\text{Amplitude} = 3.4 - 2.05 = 1.35
\]

### Step 2: **Period**
The period is the distance required for one complete cycle. From the graph, the function goes from 1.2 (maximum) to 3.4 (next maximum). Therefore, the period is:
\[
\text{Period} = 3.4 - 1.2 = 2.2
\]
The period of a cosine function is given by \( \frac{2\pi}{B} \), where \(B\) is the frequency. Setting the period equal to 2.2 gives:
\[
\frac{2\pi}{B} = 2.2 \quad \Rightarrow \quad B = \frac{2\pi}{2.2} = \frac{10\pi}{11}
\]

### Step 3: **Phase Shift**
The phase shift occurs when the graph is shifted horizontally. Since the graph reaches a maximum at \(x = 1.2\), and a cosine function normally has a maximum at \(x = 0\), the phase shift is:
\[
\text{Phase Shift} = 1.2
\]

### Step 4: **Vertical Shift**
The vertical shift is the midline, which we already calculated as 2.05.

### Formula
Combining all these components, we get the function:
\[
y = 1.35 \cos\left(\frac{10\pi}{11}(x - 1.2)\right) + 2.05
\]

Would you like further details on any step or have any questions?

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Here are some related questions to consider:
1. How would the formula change if the graph represented a sine wave instead of cosine?
2. What if the vertical shift was negative? How would the formula adjust?
3. How can we determine the phase shift if the graph starts at a minimum?
4. What is the relationship between the period and frequency of a trigonometric function?
5. How would stretching or compressing the graph vertically affect the amplitude?

**Tip:** Always ensure the values of maximum and minimum points are accurately noted to avoid errors when determining the amplitude and vertical shift.

Find a formula for the given function based on the graph showing a cosine wave.

Learn how to derive the cosine function formula from a graph. This step-by-step solution covers calculating amplitude, period, phase shift, and vertical shift for the given cosine function, suitable for students in Grades 10-12.

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