The function given is:

$y = \sin\left(x + \frac{\pi}{10}\right)$

We can now find the amplitude, period, vertical shift, and phase shift of the function:

1. Amplitude:
   The amplitude of a sine function $y = \sin(bx + c)$ is given by the coefficient of the sine function. Since there is no coefficient in front of the sine function here, the amplitude is 1.

2. Period:
   The period of a sine function is determined by the formula:

   $\text{Period} = \frac{2\pi}{|b|}$

   In this case, $b = 1$ (since the function is $\sin(x + \frac{\pi}{10})$), so the period is:

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

3. Vertical Shift:
   The vertical shift occurs when there is an added constant outside the sine function. Since there is no constant added outside, the vertical shift is 0.

4. Phase Shift:
   The phase shift is determined by the horizontal translation of the sine wave and is calculated using:

   $\text{Phase Shift} = \frac{-c}{b}$

   Here, $c = \frac{\pi}{10}$ and $b = 1$, so the phase shift is:

   $\text{Phase Shift} = \frac{-\frac{\pi}{10}}{1} = -\frac{\pi}{10}$

   Therefore, the graph is shifted to the left by $\frac{\pi}{10}$.

Summary:

• Amplitude: 1
• Period: $2\pi$
• Vertical Shift: 0
• Phase Shift: $-\frac{\pi}{10}$ (left by $\frac{\pi}{10}$)

Would you like further details on any part of this explanation?

Here are five related questions:

1. How does a coefficient in front of the sine function affect its amplitude?
2. How would the period change if the function was $y = \sin(2x + \frac{\pi}{10})$?
3. What happens to the graph if we introduce a vertical shift?
4. How would you graph the function using the phase shift information?
5. What effect would multiplying the sine function by a negative value have on its graph?

Tip: To find the phase shift, always check the sign inside the parentheses with $x$ for proper direction of shift.