The function provided in the image is:

$y = -1 + 5 \sin\left(\frac{x + \pi}{2}\right)$

Let's break down the components:

1. Vertical Shift: The vertical shift is determined by the constant term added or subtracted from the function. In this case, it's -1, which means the entire sine wave is shifted down by 1 unit.

2. Amplitude: The amplitude is the coefficient of the sine function. In this case, it's 5, meaning the sine wave oscillates 5 units above and below the midline.

3. Period: The period of a 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, the coefficient of $x$ is $\frac{1}{2}$, so the period is:

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

   Therefore, the period is $4\pi$.

4. Phase Shift: The phase shift is determined by the horizontal translation, which is affected by $\frac{x + \pi}{2}$. To find the phase shift, set the argument of the sine function to zero:

   $\frac{x + \pi}{2} = 0 \quad \Rightarrow \quad x = -\pi$

   So, the phase shift is $-\pi$, meaning the graph is shifted left by $\pi$ units.

Summary:

• Vertical Shift: Down 1 unit
• Amplitude: 5
• Period: $4\pi$
• Phase Shift: Left $\pi$ units

Would you like more details on graphing or any other aspect? Here are some follow-up questions:

1. How do phase shifts affect the appearance of sine and cosine graphs?
2. What would the graph look like if the amplitude were negative?
3. How can we determine the frequency of this sine wave from the period?
4. What is the effect of adding a vertical stretch to a sine function?
5. How would the graph change if we altered the vertical shift to +3?

Tip: Always visualize the midline and the amplitude together when graphing vertical shifts.