Fill out the following information to write a $$ -\sin $$ equation: Middle $$ =3 $$ Amplitude $$ =1 $$ Period $$ =4 \pi $$ Shift to the left $$ =-3 \pi $$ Write $$ \mathrm{a}-\sin $$ equation $$ y=3-1 \sin \left(\frac{1}{2} x+3 \pi ight) imes $$ *his graph has a nonzero shift.

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Fill out the following information to write a $$ -\sin $$ equation: Middle $$ =3 $$ Amplitude $$ =1 $$ Period $$ =4 \pi $$ Shift to the left $$ =-3 \pi $$ Write $$ \mathrm{a}-\sin $$ equation $$ y=3-1 \sin \left(\frac{1}{2} x+3 \piight) 	imes $$ *his graph has a nonzero shift.

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To write the equation of a sine function with the given parameters, we can follow the general form of a sine function:

$$
y = D + A \sin(B(x - C))
$$

Where:
- $$ D $$ is the vertical shift (middle value).
- $$ A $$ is the amplitude.
- $$ B $$ is related to the period.
- $$ C $$ is the horizontal shift.

### Given Information:
- Middle $$ D = 3 $$
- Amplitude $$ A = 1 $$
- Period $$ P = 4\pi $$
- Shift to the left $$ C = -3\pi $$

### Step 1: Determine $$ B $$
The period $$ P $$ is related to $$ B $$ by the formula:

$$
P = \frac{2\pi}{B}
$$

We can rearrange this to find $$ B $$:

$$
B = \frac{2\pi}{P} = \frac{2\pi}{4\pi} = \frac{1}{2}
$$

### Step 2: Write the Equation
Now we can substitute the values into the sine function equation:

$$
y = D + A \sin(B(x - C))
$$

Substituting the values we have:

$$
y = 3 + 1 \sin\left(\frac{1}{2}(x + 3\pi)\right)
$$

### Final Equation
Thus, the equation can be written as:

$$
y = 3 - \sin\left(\frac{1}{2}x + \frac{3\pi}{2}\right)
$$

This accounts for the negative sine function, which reflects the graph across the horizontal axis.

So the final equation is:

$$
y = 3 - \sin\left(\frac{1}{2}x + 3\pi\right)
$$
The equation is $$ y = 3 - \sin\left(\frac{1}{2}x + 3\pi\right) $$.

Fill out the following information to write a \( -\sin \) equation: Middle \( =3 \) Amplitude \( =1 \) Period \( =4 \pi \) Shift to the left \( =-3 \pi \) Write \( \mathrm{a}-\sin \) equation \( y=3-1 \sin \left(\frac{1}{2} x+3 \pi\right) \times \) *his graph has a nonzero shift.

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