46.) $$ y=\sin \frac{\pi x}{4} $$

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The function $$ y = \sin \frac{\pi x}{4} $$ is a sine function that has been transformed from the basic sine function $$ y = \sin x $$. Let's analyze its properties:

1. **Amplitude**: The amplitude of the sine function is the coefficient in front of the sine. In this case, the amplitude is 1 (since there is no coefficient other than 1).

2. **Period**: The period of the sine function is given by the formula:
   $$
   \text{Period} = \frac{2\pi}{b}
   $$
   where $$ b $$ is the coefficient of $$ x $$ inside the sine function. Here, $$ b = \frac{\pi}{4} $$, so the period is:
   $$
   \text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8
   $$

3. **Phase Shift**: There is no horizontal shift in this function since there is no addition or subtraction inside the argument of the sine function.

4. **Vertical Shift**: There is also no vertical shift since there is no constant added or subtracted from the sine function.

5. **Key Points**: The sine function typically has key points at:
   - $$ (0, 0) $$
   - $$ \left(\frac{1}{4} \text{Period}, 1\right) $$
   - $$ \left(\frac{1}{2} \text{Period}, 0\right) $$
   - $$ \left(\frac{3}{4} \text{Period}, -1\right) $$
   - $$ \left(\text{Period}, 0\right) $$

For this function, the key points will occur at:
   - $$ (0, 0) $$
   - $$ (2, 1) $$ (since $$ \frac{1}{4} \times 8 = 2 $$)
   - $$ (4, 0) $$ (since $$ \frac{1}{2} \times 8 = 4 $$)
   - $$ (6, -1) $$ (since $$ \frac{3}{4} \times 8 = 6 $$)
   - $$ (8, 0) $$ (since $$ 8 $$ is the period)

6. **Graph**: The graph of $$ y = \sin \frac{\pi x}{4} $$ will oscillate between 1 and -1, completing one full cycle from $$ x = 0 $$ to $$ x = 8 $$.

In summary, the function $$ y = \sin \frac{\pi x}{4} $$ has an amplitude of 1, a period of 8, and key points at $$ (0, 0) $$, $$ (2, 1) $$, $$ (4, 0) $$, $$ (6, -1) $$, and $$ (8, 0) $$.
The function $$ y = \sin \frac{\pi x}{4} $$ has an amplitude of 1, a period of 8, and key points at $$ (0, 0) $$, $$ (2, 1) $$, $$ (4, 0) $$, $$ (6, -1) $$, and $$ (8, 0) $$.

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