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To derive the equation of the cosine curve based on the given properties:

1. **Amplitude**: The amplitude is \(3\), so the coefficient of cosine is \(3\).
2. **Period**: The period is \(3\pi\). The formula for the period is given by:
   \[
   \text{Period} = \frac{2\pi}{B}
   \]
   Solving for \(B\):
   \[
   B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{3\pi} = \frac{2}{3}
   \]
3. **Vertical Shift**: The equation of the axis is \(y = 2\), so the graph is shifted up by \(2\).
4. **Horizontal Shift**: The curve is shifted \(\frac{\pi}{4}\) radians to the left, so the phase shift is \(+\frac{\pi}{4}\).

The general form of a cosine function is:
\[
y = A\cos(B(x - C)) + D
\]
where:
- \(A\) is the amplitude (\(3\)),
- \(B\) is the angular frequency (\(\frac{2}{3}\)),
- \(C\) is the horizontal shift (\(-\frac{\pi}{4}\), becomes \(+\frac{\pi}{4}\) in the equation),
- \(D\) is the vertical shift (\(2\)).

Substituting these values, the equation becomes:
\[
y = 3\cos\left(\frac{2}{3}\left(x + \frac{\pi}{4}\right)\right) + 2
\]

### Final Equation:
\[
y = 3\cos\left(\frac{2}{3}x + \frac{\pi}{6}\right) + 2
\]

Let me know if you'd like a detailed explanation of any step or further assistance.

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### Related Questions:
1. How does the phase shift affect the cosine function's graph?
2. Can you derive the equation for a sine curve with the same properties?
3. How would the equation change if the period were \(4\pi\) instead of \(3\pi\)?
4. What happens to the graph if the amplitude is doubled?
5. Can you explain the relationship between angular frequency (\(B\)) and period?

### Tip:
Always verify the period by recalculating using \(B = \frac{2\pi}{\text{Period}}\) to avoid mistakes in transformations.

A cosine curve has an amplitude of 3 units and a period of 3π radians. The equation of the axis is y = 2, and a horizontal shift of π/4 radians to the left has been applied. Write the equation of this function.

Learn how to derive the equation of a cosine function with an amplitude of 3, a period of 3π radians, a vertical shift of 2, and a horizontal shift of π/4 radians to the left. This guide provides a step-by-step solution suitable for high school students in Grades 10-12.

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