Matching trigonometric equations to their corresponding sine and cosine graphs.

Problem Overview:

We are tasked with matching each given equation of a sine or cosine function to its corresponding graph. For each equation, we need to determine the amplitude, period, phase shift, and vertical shift (if any). Then, we will match the equations to the graphs provided.

Key Properties of Sine and Cosine Functions:

1. General Form:
- Sine: \( y = A \sin(Bx - C) + D \)
- Cosine: \( y = A \cos(Bx - C) + D \)
2. Amplitude: \( |A| \)
3. Period: \( \frac{2\pi}{|B|} \)
4. Phase Shift: \( \frac{C}{B} \) (to the right if positive, to the left if negative)
5. Vertical Shift: \( D \)

Step-by-Step Solution:

#### Equation P: \( y = 2 \sin x + \frac{1}{2} \)
- Amplitude: \( |2| = 2 \)
- Period: \( \frac{2\pi}{1} = 2\pi \)
- Phase Shift: \( 0 \) (no phase shift)
- Vertical Shift: \( \frac{1}{2} \) (upward)
- Graph Characteristics: The graph is a sine wave with amplitude 2, period \( 2\pi \), and shifted up by \( \frac{1}{2} \).
- Matching Graph: Graph 3

#### Equation R: \( y = -2 \cos \left( \frac{x - \pi}{2} \right) \)
- Amplitude: \( |-2| = 2 \)
- Period: \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)
- Phase Shift: \( \frac{\pi}{\frac{1}{2}} = 2\pi \) (to the right)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a cosine wave with amplitude 2, period \( 4\pi \), phase shift \( 2\pi \) to the right, and reflected over the x-axis.
- Matching Graph: Graph 5

#### Equation N: \( y = 2 \sin \left( 2 \left( x + \frac{\pi}{2} \right) \right) + 2 \)
- Amplitude: \( |2| = 2 \)
- Period: \( \frac{2\pi}{2} = \pi \)
- Phase Shift: \( -\frac{\pi}{2} \) (to the left)
- Vertical Shift: \( 2 \) (upward)
- Graph Characteristics: The graph is a sine wave with amplitude 2, period \( \pi \), phase shift \( \frac{\pi}{2} \) to the left, and shifted up by 2.
- Matching Graph: Graph 2

#### Equation H: \( y = -2 \sin \left( x - \frac{\pi}{2} \right) \)
- Amplitude: \( |-2| = 2 \)
- Period: \( \frac{2\pi}{1} = 2\pi \)
- Phase Shift: \( \frac{\pi}{2} \) (to the right)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a sine wave with amplitude 2, period \( 2\pi \), phase shift \( \frac{\pi}{2} \) to the right, and reflected over the x-axis.
- Matching Graph: Graph 4

#### Equation M: \( y = -2 \cos \left( \frac{x + \pi}{2} \right) \)
- Amplitude: \( |-2| = 2 \)
- Period: \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)
- Phase Shift: \( -\pi \) (to the left)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a cosine wave with amplitude 2, period \( 4\pi \), phase shift \( \pi \) to the left, and reflected over the x-axis.
- Matching Graph: Graph 1

#### Equation L: \( y = 2 \sin \left( 2 \left( x + \frac{\pi}{4} \right) \right) \)
- Amplitude: \( |2| = 2 \)
- Period: \( \frac{2\pi}{2} = \pi \)
- Phase Shift: \( -\frac{\pi}{4} \) (to the left)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a sine wave with amplitude 2, period \( \pi \), and phase shift \( \frac{\pi}{4} \) to the left.
- Matching Graph: Graph 2 (already matched, so this is incorrect. Re-evaluate.)

#### Equation D: \( y = \cos \left( 2 \left( x + \frac{\pi}{4} \right) \right) + 2 \)
- Amplitude: \( |1| = 1 \)
- Period: \( \frac{2\pi}{2} = \pi \)
- Phase Shift: \( -\frac{\pi}{4} \) (to the left)
- Vertical Shift: \( 2 \) (upward)
- Graph Characteristics: The graph is a cosine wave with amplitude 1, period \( \pi \), phase shift \( \frac{\pi}{4} \) to the left, and shifted up by 2.
- Matching Graph: Graph 2 (already matched, so this is incorrect. Re-evaluate.)

#### Equation E: \( y = -2 \sin \left( \frac{1}{2} (x + 2\pi) \right) \)
- Amplitude: \( |-2| = 2 \)
- Period: \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)
- Phase Shift: \( -2\pi \) (to the left)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a sine wave with amplitude 2, period \( 4\pi \), phase shift \( 2\pi \) to the left, and reflected over the x-axis.
- Matching Graph: Graph 5 (already matched, so this is incorrect. Re-evaluate.)

#### Equation O: \( y = -2 \sin x \)
- Amplitude: \( |-2| = 2 \)
- Period: \( \frac{2\pi}{1} = 2\pi \)
- Phase Shift: \( 0 \)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a sine wave with amplitude 2, period \( 2\pi \), and reflected over the x-axis.
- Matching Graph: Graph 4 (already matched, so this is incorrect. Re-evaluate.)

#### Equation A: \( y = 2 \cos \left( x - \frac{\pi}{2} \right) + \frac{1}{2} \)
- Amplitude: \( |2| = 2 \)
- Period: \( \frac{2\pi}{1} = 2\pi \)
- Phase Shift: \( \frac{\pi}{2} \) (to the right)
- Vertical Shift: \( \frac{1}{2} \) (upward)
- Graph Characteristics: The graph is a cosine wave with amplitude 2, period \( 2\pi \), phase shift \( \frac{\pi}{2} \) to the right, and shifted up by \( \frac{1}{2} \).
- Matching Graph: Graph 3 (already matched, so this is incorrect. Re-evaluate.)

#### Equation T: \( y = -2 \cos \left( \frac{1}{2} (x + 2\pi) \right) \)
- Amplitude: \( |-2| = 2 \)
- Period: \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)
- Phase Shift: \( -2\pi \) (to the left)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a cosine wave with amplitude 2, period \( 4\pi \), phase shift \( 2\pi \) to the left, and reflected over the x-axis.
- Matching Graph: Graph 5 (already matched, so this is incorrect. Re-evaluate.)

#### Equation I: \( y = 2 \cos 2x \)
- Amplitude: \( |2| = 2 \)
- Period: \( \frac{2\pi}{2} = \pi \)
- Phase Shift: \( 0 \)
- Vertical Shift: \( 0 \)
- Graph Characteristics: The graph is a cosine wave with amplitude 2, period \( \pi \), and no phase shift.
- Matching Graph: Graph 2 (already matched, so this is incorrect. Re-evaluate.)

Final Matches:

1. \( M \): \( y = -2 \cos \left( \frac{x + \pi}{2} \right) \) → Graph 1
2. \( N \): \( y = 2 \sin \left( 2 \left( x + \frac{\pi}{2} \right) \right) + 2 \) → Graph 2
3. \( P \): \( y = 2 \sin x + \frac{1}{2} \) → Graph 3
4. \( H \): \( y = -2 \sin \left( x - \frac{\pi}{2} \right) \) → Graph 4
5. \( R \): \( y = -2 \cos \left( \frac{x - \pi}{2} \right) \) → Graph 5

Final Answer:

\[
\boxed{M, N, P, H, R}
\]