To find the graph of the function $2 \sin \left( \frac{1}{2}x - \frac{\pi}{6} \right)$ with a period of $4\pi$, we need to first understand how the period and the transformations affect the graph.

Step 1: Determine the period of the function

The standard sine function $y = \sin(x)$ has a period of $2\pi$. The formula for the period of a sine function is:

$\text{Period} = \frac{2\pi}{\left|\text{coefficient of } x\right|}$

In our case, the coefficient of $x$ inside the sine function is $\frac{1}{2}$, so the period becomes:

$\text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi$

Thus, the period is indeed $4\pi$, as specified in the problem.

Step 2: Transformations

• Amplitude: The amplitude is $2$, so the graph will oscillate between $-2$ and $2$.
• Phase shift: The term $-\frac{\pi}{6}$ indicates a phase shift of $\frac{\pi}{6}$ units to the right.

Step 3: Graphing the function

Given that the period is $4\pi$, the graph completes one full oscillation over the interval $[0, 4\pi]$.

To sketch the graph, we divide the period into critical points:

• The function will complete a full cycle over the interval $[0, 4\pi]$.
• Key points to consider are the start and end points of the period: $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$, $\frac{5\pi}{2}$, $3\pi$, and $4\pi$.

However, since the question asks for which values of $x$ to graph, the options are:

1. $\pi$, $2\pi$, $3\pi$, $4\pi$ — This would be incorrect because it only covers part of the cycle (from $\pi$ to $4\pi$).
2. $\frac{\pi}{3}$, $\frac{2\pi}{3}$, $\pi$, $\frac{4\pi}{3}$ — This is also incorrect as it only covers part of the function.
3. The correct answer is: $0$, $\pi$, $2\pi$, $3\pi$, and $4\pi$.

These are the critical points of one full cycle of the sine function with the given period and phase shift.

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Would you like a graph of this function for better visualization, or do you have any more questions on this? Also, here are some related questions:

1. How do we calculate the amplitude and period of a sine wave?
2. What does a phase shift do to the graph of a sine function?
3. How can we graph transformations of trigonometric functions systematically?
4. What is the difference between a phase shift and a vertical shift in sine functions?
5. How would the graph change if the period were $2\pi$ instead of $4\pi$?

Tip: Remember that a phase shift moves the entire graph left or right but does not affect the period or amplitude!