Let's solve the given problems step by step.

1. Equation of the cosine function

• Amplitude = 3
• Period = $3\pi$, so the angular frequency $b$ is: $b = \frac{2\pi}{\text{Period}} = \frac{2\pi}{3\pi} = \frac{2}{3}$
• The axis is $y = 2$, which is the vertical shift.
• The function is shifted $\frac{\pi}{4}$ radians to the left, which is represented by $x + \frac{\pi}{4}$.

The general form of a cosine function is: $y = A \cos\left(b(x - h)\right) + k$

Substituting the given values: $y = 3 \cos\left(\frac{2}{3}\left(x + \frac{\pi}{4}\right)\right) + 2$

This is the equation of the cosine function.

2. Values of the function for specific $x$ values

Substituting $x = \frac{\pi}{2}$:

$y = 3 \cos\left(\frac{2}{3}\left(\frac{\pi}{2} + \frac{\pi}{4}\right)\right) + 2$ Simplify inside the cosine: $\frac{\pi}{2} + \frac{\pi}{4} = \frac{2\pi}{4} + \frac{\pi}{4} = \frac{3\pi}{4}$ $y = 3 \cos\left(\frac{2}{3} \cdot \frac{3\pi}{4}\right) + 2$ Simplify further: $\frac{2}{3} \cdot \frac{3\pi}{4} = \frac{6\pi}{12} = \frac{\pi}{2}$ $y = 3 \cos\left(\frac{\pi}{2}\right) + 2$ Since $\cos\left(\frac{\pi}{2}\right) = 0$: $y = 3(0) + 2 = 2$

Substituting $x = \frac{3\pi}{4}$:

$y = 3 \cos\left(\frac{2}{3}\left(\frac{3\pi}{4} + \frac{\pi}{4}\right)\right) + 2$ Simplify inside the cosine: $\frac{3\pi}{4} + \frac{\pi}{4} = \frac{4\pi}{4} = \pi$ $y = 3 \cos\left(\frac{2}{3} \cdot \pi\right) + 2$ Simplify further: $\frac{2}{3} \cdot \pi = \frac{2\pi}{3}$ $y = 3 \cos\left(\frac{2\pi}{3}\right) + 2$ Since $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$: $y = 3\left(-\frac{1}{2}\right) + 2 = -\frac{3}{2} + 2 = \frac{1}{2}$

Substituting $x = \frac{11\pi}{6}$:

$y = 3 \cos\left(\frac{2}{3}\left(\frac{11\pi}{6} + \frac{\pi}{4}\right)\right) + 2$ Simplify inside the cosine: $\frac{11\pi}{6} + \frac{\pi}{4} = \frac{22\pi}{12} + \frac{3\pi}{12} = \frac{25\pi}{12}$ $y = 3 \cos\left(\frac{2}{3} \cdot \frac{25\pi}{12}\right) + 2$ Simplify further: $\frac{2}{3} \cdot \frac{25\pi}{12} = \frac{50\pi}{36} = \frac{25\pi}{18}$ So: $y = 3 \cos\left(\frac{25\pi}{18}\right) + 2$ This requires approximation: $\cos\left(\frac{25\pi}{18}\right) \approx -0.866$ $y = 3(-0.866) + 2 \approx -2.598 + 2 = -0.598$

Final Results:

1. Equation: $y = 3 \cos\left(\frac{2}{3}\left(x + \frac{\pi}{4}\right)\right) + 2$
2. Values:
   • $x = \frac{\pi}{2}$: $y = 2$
   • $x = \frac{3\pi}{4}$: $y = \frac{1}{2}$
   • $x = \frac{11\pi}{6}$: $y \approx -0.598$

Do you want more details or have further questions? Here are 5 related queries:

1. What is the difference between amplitude and vertical shift in trigonometric functions?
2. How do horizontal shifts affect the phase of a cosine or sine function?
3. What is the importance of the period in a trigonometric function?
4. How would the function change if the amplitude were halved?
5. Can we solve the values using an alternate method, like a unit circle?

Tip: Always double-check angles and simplifications when substituting values into trigonometric functions!