Algebra 2 worksheet on graphing trigonometric functions, including sine, cosine, and their transformations with amplitude and period calculations.

Let’s solve each of the 6 trigonometric graphing problems step by step. For each, we’ll find the amplitude and period (where applicable), then describe how to graph it.

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🔹 General Form Reminder:

For functions of the form:

- y = A sin(Bθ + C) + D or y = A cos(Bθ + C) + D
- Amplitude = |A| → vertical stretch/compression
- Period = 2π / |B| → horizontal stretch/compression
- Phase Shift = -C/B (not needed here)
- Vertical Shift = D → moves graph up/down

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## ✔ Problem 1: y = 2 sin θ

- Amplitude = |2| = 2
- Period = 2π / 1 = 2π
- Graphing: Standard sine wave, but stretched vertically from -2 to 2.
- Key points over [0, 2π]:
- θ=0 → y=0
- θ=π/2 → y=2
- θ=π → y=0
- θ=3π/2 → y=-2
- θ=2π → y=0

✔ Plot these points and connect with a smooth wave.

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## ✔ Problem 2: y = (1/2) cos θ

- Amplitude = |1/2| = 0.5
- Period = 2π / 1 = 2π
- Graphing: Standard cosine wave, compressed vertically from -0.5 to 0.5.
- Key points over [0, 2π]:
- θ=0 → y=0.5
- θ=π/2 → y=0
- θ=π → y=-0.5
- θ=3π/2 → y=0
- θ=2π → y=0.5

✔ Plot these and draw a smooth cosine curve.

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## ✔ Problem 3: y = 2 sin(θ/2)

- Amplitude = |2| = 2
- Period = 2π / |B| = 2π / (1/2) = 4π
- Graphing: Sine wave stretched vertically (±2) and horizontally (period = 4π).
- Key points over [0, 4π]:
- θ=0 → y=0
- θ=π → y=2 (since θ/2 = π/2 → sin(π/2)=1 → y=2×1=2)
- θ=2π → y=0
- θ=3π → y=-2
- θ=4π → y=0

✔ Plot over [0, 4π] — note the x-axis goes to 6π, so you can plot one full cycle.

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## ✔ Problem 4: y = cos(3θ)

- Amplitude = |1| = 1
- Period = 2π / |3| = 2π/3
- Graphing: Cosine wave compressed horizontally — completes 3 cycles in 2π.
- One period: [0, 2π/3]
- θ=0 → y=1
- θ=π/6 → y=cos(π/2)=0
- θ=π/3 → y=cos(π)=-1
- θ=π/2 → y=cos(3π/2)=0
- θ=2π/3 → y=cos(2π)=1

✔ You can plot 3 full cycles between 0 and 2π.

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## ✔ Problem 5: y = sin θ - 1

- Amplitude = |1| = 1
- Period = 2π / 1 = 2π
- Vertical Shift = -1 → entire graph shifted down 1 unit
- Range: from -1 -1 = -2 to 1 -1 = 0
- Graphing:
- Standard sine wave, but shifted down.
- Key points over [0, 2π]:
- θ=0 → y=0 -1 = -1
- θ=π/2 → y=1 -1 = 0
- θ=π → y=0 -1 = -1
- θ=3π/2 → y=-1 -1 = -2
- θ=2π → y=0 -1 = -1

✔ Plot and connect — now centered at y = -1.

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## ✔ Problem 6: y = 2 + cos θ

- Amplitude = |1| = 1
- Period = 2π / 1 = 2π
- Vertical Shift = +2 → entire graph shifted up 2 units
- Range: from -1 + 2 = 1 to 1 + 2 = 3
- Graphing:
- Standard cosine wave, shifted up.
- Key points over [0, 2π]:
- θ=0 → y=1 + 2 = 3
- θ=π/2 → y=0 + 2 = 2
- θ=π → y=-1 + 2 = 1
- θ=3π/2 → y=0 + 2 = 2
- θ=2π → y=1 + 2 = 3

✔ Plot and connect — now centered at y = 2.

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## 📊 Summary Table:

| Function | Amplitude | Period | Vertical Shift | Notes |
|------------------|-----------|----------|----------------|--------------------------------|
| 1) y = 2 sin θ | 2 | 2π | 0 | Tall sine wave |
| 2) y = (1/2) cos θ | 0.5 | 2π | 0 | Short cosine wave |
| 3) y = 2 sin(θ/2) | 2 | 4π | 0 | Long-period sine |
| 4) y = cos(3θ) | 1 | 2π/3 | 0 | Fast cosine (3 cycles in 2π) |
| 5) y = sin θ - 1 | 1 | 2π | -1 | Sine shifted down |
| 6) y = 2 + cos θ | 1 | 2π | +2 | Cosine shifted up |

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## 🖍️ Graphing Tips for Students:

- Use the key points (max, min, zeros) to sketch accurately.
- Label your axes clearly — especially radians on x-axis.
- For functions with period ≠ 2π, mark the new period on the x-axis.
- For vertical shifts, draw a dashed line at y = D as the “centerline.”

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✔ All problems solved! You can now confidently graph each function using the amplitude, period, and vertical shift.

Let me know if you’d like hand-drawn sketches or printable graphs!