To graph the function $y = \sin\left(\frac{1}{2}x + \frac{\pi}{6}\right)$, let's break down the important elements and then proceed with graphing.

1. Base Function:

The base function is $\sin(x)$, which is a periodic function with a period of $2\pi$ and amplitude 1. The sine function oscillates between -1 and 1.

2. Horizontal Stretching:

The coefficient $\frac{1}{2}$ inside the sine function causes a horizontal stretch. Normally, the period of $\sin(x)$ is $2\pi$. To find the new period, we divide the standard period by the coefficient of $x$: $\text{New Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi$ This means the graph will take twice as long to complete one full cycle.

3. Horizontal Shift:

The $+ \frac{\pi}{6}$ inside the function causes a horizontal shift. To determine the shift, we solve: $x = -\frac{\pi}{6}$ This indicates that the entire graph is shifted to the left by $\frac{\pi}{6}$.

4. Amplitude:

The amplitude is not affected by any coefficient outside the sine function, so it remains 1 (the graph oscillates between -1 and 1).

Step-by-Step Graphing:

1. Start with $y = \sin(x)$, as the base graph.
2. Apply the horizontal stretch: Compress the graph horizontally so that the period becomes $4\pi$.
3. Apply the horizontal shift: Shift the entire graph to the left by $\frac{\pi}{6}$.
4. The amplitude remains 1, so the graph oscillates between -1 and 1.

Appropriate Starting Graph:

A good starting graph would be $y = \sin(x)$, then apply the stretch and shift as described above. The graph will complete one full period between $x = -\frac{\pi}{6}$ and $x = \frac{11\pi}{6}$.

Would you like me to provide a sketch or simulation of the graph?

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Here are some related questions:

1. How does the horizontal stretch affect the period of sine functions?
2. What is the effect of adding $\frac{\pi}{6}$ inside the sine function?
3. How do you determine amplitude and period changes for sine graphs?
4. How can vertical shifts affect sine and cosine graphs?
5. What would happen if the coefficient in front of $x$ were negative?

Tip: When graphing trigonometric functions, always start with the base function and apply transformations step by step!