Honors Algebra 2 worksheet focusing on sine and cosine graphs, requiring students to determine transformations and graph each function.

Let’s solve each problem step by step. We’ll find the period, amplitude, reflection, phase shift, vertical shift, and list transformations in order for each sine function.

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Problem 1: y = (2/3) sin x

- Period: The parent function y = sin x has period 2π. Since there’s no coefficient inside the sine (like sin(bx)), b = 1 → Period = 2π / 1 = 2π
- Amplitude: The number in front is 2/3 → Amplitude = 2/3
- Reflection: No negative sign in front → none
- Phase Shift: No horizontal shift (no “x - c” inside) → none
- Vertical Shift: No constant added or subtracted at the end → none
- Transformations: Only one transformation — vertical compression by factor of 2/3. So:
→ *Vertical compression by 2/3*

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Problem 2: y = 4 sin(πx)

- Period: Here, b = π → Period = 2π / π = 2
- Amplitude: Coefficient in front is 4 → Amplitude = 4
- Reflection: No negative sign → none
- Phase Shift: Nothing added/subtracted to x inside → none
- Vertical Shift: No constant at the end → none
- Transformations: First, change the period (horizontal compression), then stretch vertically. Order matters!
→ *Horizontal compression so period becomes 2, then vertical stretch by 4*

Wait — let’s be precise about order. According to instructions: write transformations IN THE ORDER THEY OCCUR.

When you have y = A sin(Bx), you first apply the horizontal scaling (which affects period), then vertical scaling.

So:
→ *Horizontal compression by factor of 1/π (to get period 2), then vertical stretch by 4*

But since we usually describe it as “change period first”, and then amplitude, that’s fine.

Actually, standard order: horizontal changes first (inside the function), then vertical (outside). So:

→ *Change period to 2 (by compressing horizontally), then stretch vertically by 4*

We can say:
→ *Period changed to 2, then amplitude scaled to 4*

But to match worksheet style:
→ *Horizontal compression (period = 2), vertical stretch (amplitude = 4)*

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Problem 3: y = 2 sin(½x)

- Period: b = ½ → Period = 2π / (½) = 4π → 4π
- Amplitude: 2 → 2
- Reflection: No negative → none
- Phase Shift: none → none
- Vertical Shift: none → none
- Transformations: Again, horizontal change first (stretch because b < 1), then vertical stretch.
→ *Horizontal stretch so period becomes 4π, then vertical stretch by 2*

Or:
→ *Period changed to 4π, then amplitude scaled to 2*

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Problem 4: y = sin x + 3

- Period: b = 1 → Period = 2π → 2π
- Amplitude: coefficient is 1 → 1
- Reflection: no negative → none
- Phase Shift: none → none
- Vertical Shift: +3 at the end → up 3
- Transformations: Only vertical shift happens here.
→ *Shift graph up 3 units*

Note: Even though amplitude is 1, we still write it. And since no other changes, only vertical shift.

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Now, double-check all answers:

✔ Problem 1: All correct — just amplitude changed.

✔ Problem 2: Period = 2π/π = 2 ✔️, Amp = 4 ✔️, no shifts ✔️

✔ Problem 3: Period = 2π/(0.5) = 4π ✔️, Amp = 2 ✔️

✔ Problem 4: Vertical shift up 3 ✔️, everything else same as parent ✔️

All good!

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Final Answer:

1. y = (2/3) sin x
Period: 2π
Amplitude: 2/3
Reflection: none
Phase Shift: none
Vertical Shift: none
Transformations: Vertical compression by 2/3

2. y = 4 sin(πx)
Period: 2
Amplitude: 4
Reflection: none
Phase Shift: none
Vertical Shift: none
Transformations: Horizontal compression (period = 2), then vertical stretch by 4

3. y = 2 sin(½x)
Period: 4π
Amplitude: 2
Reflection: none
Phase Shift: none
Vertical Shift: none
Transformations: Horizontal stretch (period = 4π), then vertical stretch by 2

4. y = sin x + 3
Period: 2π
Amplitude: 1
Reflection: none
Phase Shift: none
Vertical Shift: up 3
Transformations: Shift graph up 3 units