Writing equations from graphs of trigonometric functions.

To solve the problem, we need to analyze each graph and determine the amplitude, period, phase shift, vertical shift, equation of the midline, and the sine or cosine equation that represents the graph. Let's go through each graph step by step.

---

Graph 1:

#### Observations:
- The graph is a sine wave.
- The amplitude is the maximum distance from the midline to the peak or trough. Here, it appears to be 2.
- The period is the length of one complete cycle. It looks like the period is \(2\pi\).
- There is no visible phase shift (horizontal shift).
- There is no vertical shift; the midline appears to be the x-axis (y = 0).

#### Calculations:
1. Amplitude (A): The maximum value is 2, so \(A = 2\).
2. Period: The period is \(2\pi\). For a sine function, the period is given by \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\). Since the period is \(2\pi\), \(B = 1\).
3. Phase Shift (C): There is no phase shift, so \(C = 0\).
4. Vertical Shift (D): The midline is at \(y = 0\), so \(D = 0\).
5. Equation of the Midline: \(y = 0\).
6. Sine Equation: The general form is \(y = A \sin(Bx - C) + D\). Substituting the values, we get:
\[
y = 2 \sin(x)
\]

#### Final Answers for Graph 1:
- Amplitude: 2
- Period: \(2\pi\)
- Phase Shift: 0
- Vertical Shift: 0
- Equation of the Midline: \(y = 0\)
- Sine Equation: \(y = 2 \sin(x)\)

---

Graph 2:

#### Observations:
- The graph is a cosine wave.
- The amplitude is 1.
- The period is \(4\pi\).
- There is no visible phase shift.
- There is no vertical shift; the midline appears to be the x-axis (y = 0).

#### Calculations:
1. Amplitude (A): The maximum value is 1, so \(A = 1\).
2. Period: The period is \(4\pi\). For a cosine function, the period is given by \(\frac{2\pi}{B}\). Solving for \(B\):
\[
4\pi = \frac{2\pi}{B} \implies B = \frac{1}{2}
\]
3. Phase Shift (C): There is no phase shift, so \(C = 0\).
4. Vertical Shift (D): The midline is at \(y = 0\), so \(D = 0\).
5. Equation of the Midline: \(y = 0\).
6. Cosine Equation: The general form is \(y = A \cos(Bx - C) + D\). Substituting the values, we get:
\[
y = \cos\left(\frac{1}{2}x\right)
\]

#### Final Answers for Graph 2:
- Amplitude: 1
- Period: \(4\pi\)
- Phase Shift: 0
- Vertical Shift: 0
- Equation of the Midline: \(y = 0\)
- Cosine Equation: \(y = \cos\left(\frac{1}{2}x\right)\)

---

Graph 3:

#### Observations:
- The graph is a sine wave.
- The amplitude is 1.
- The period is \(2\pi\).
- There is a phase shift to the right by \(\pi/2\).
- There is no vertical shift; the midline appears to be the x-axis (y = 0).

#### Calculations:
1. Amplitude (A): The maximum value is 1, so \(A = 1\).
2. Period: The period is \(2\pi\). For a sine function, the period is given by \(\frac{2\pi}{B}\). Since the period is \(2\pi\), \(B = 1\).
3. Phase Shift (C): The graph is shifted to the right by \(\pi/2\). For a sine function, the phase shift is given by \(\frac{C}{B}\). Here, \(C = \pi/2\) and \(B = 1\), so the phase shift is \(\pi/2\).
4. Vertical Shift (D): The midline is at \(y = 0\), so \(D = 0\).
5. Equation of the Midline: \(y = 0\).
6. Sine Equation: The general form is \(y = A \sin(Bx - C) + D\). Substituting the values, we get:
\[
y = \sin\left(x - \frac{\pi}{2}\right)
\]

#### Final Answers for Graph 3:
- Amplitude: 1
- Period: \(2\pi\)
- Phase Shift: \(\pi/2\) (to the right)
- Vertical Shift: 0
- Equation of the Midline: \(y = 0\)
- Sine Equation: \(y = \sin\left(x - \frac{\pi}{2}\right)\)

---

Graph 4:

#### Observations:
- The graph is a cosine wave.
- The amplitude is 1.
- The period is \(2\pi\).
- There is a vertical shift upwards by 1.
- There is no phase shift.

#### Calculations:
1. Amplitude (A): The maximum value is 2, and the minimum value is 0. The amplitude is half the distance between the maximum and minimum values:
\[
A = \frac{2 - 0}{2} = 1
\]
2. Period: The period is \(2\pi\). For a cosine function, the period is given by \(\frac{2\pi}{B}\). Since the period is \(2\pi\), \(B = 1\).
3. Phase Shift (C): There is no phase shift, so \(C = 0\).
4. Vertical Shift (D): The midline is at \(y = 1\), so \(D = 1\).
5. Equation of the Midline: \(y = 1\).
6. Cosine Equation: The general form is \(y = A \cos(Bx - C) + D\). Substituting the values, we get:
\[
y = \cos(x) + 1
\]

#### Final Answers for Graph 4:
- Amplitude: 1
- Period: \(2\pi\)
- Phase Shift: 0
- Vertical Shift: 1
- Equation of the Midline: \(y = 1\)
- Cosine Equation: \(y = \cos(x) + 1\)

---

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Graph} & \text{Amplitude} & \text{Period} & \text{Phase Shift} & \text{Vertical Shift} & \text{Equation of Midline} & \text{Equation} \\
\hline
1 & 2 & 2\pi & 0 & 0 & y = 0 & y = 2 \sin(x) \\
\hline
2 & 1 & 4\pi & 0 & 0 & y = 0 & y = \cos\left(\frac{1}{2}x\right) \\
\hline
3 & 1 & 2\pi & \pi/2 & 0 & y = 0 & y = \sin\left(x - \frac{\pi}{2}\right) \\
\hline
4 & 1 & 2\pi & 0 & 1 & y = 1 & y = \cos(x) + 1 \\
\hline
\end{array}
}
\]