I'll solve each problem step by step, working through the calculations carefully.
Problem 1: y = (2/3)sin x
Let me identify the components:
- This is in the form y = A sin(Bx - C) + D where A = 2/3, B = 1, C = 0, D = 0
Period: For sine functions, period = 2π/B = 2π/1 = 2π
Amplitude: |A| = |2/3| = 2/3
Reflection: There's no negative sign in front of the amplitude, so there's NO reflection
Phase Shift: C/B = 0/1 = 0, so there's NO phase shift
Vertical Shift: D = 0, so there's NO vertical shift
Transformations: Starting from y = sin x, we apply a vertical compression by a factor of 2/3 (multiply all y-values by 2/3).
Problem 2: y = 4 sin πx
Components: A = 4, B = π, C = 0, D = 0
Period: 2π/B = 2π/π = 2
Amplitude: |4| = 4
Reflection: No negative sign, so NO reflection
Phase Shift: 0/π = 0, so NO phase shift
Vertical Shift: D = 0, so NO vertical shift
Transformations: Starting from y = sin x, first change the period to 2 (horizontal compression), then vertically stretch by a factor of 4.
Problem 3: y = 2 sin(1/2)x
Components: A = 2, B = 1/2, C = 0, D = 0
Period: 2π/(1/2) = 2π × 2 = 4π
Amplitude: |2| = 2
Reflection: No negative sign, so NO reflection
Phase Shift: 0/(1/2) = 0, so NO phase shift
Vertical Shift: D = 0, so NO vertical shift
Transformations: Starting from y = sin x, first change the period to 4π (horizontal stretch by factor of 2), then vertically stretch by a factor of 2.
Problem 4: y = sin x + 3
Components: A = 1, B = 1, C = 0, D = 3
Period: 2π/1 = 2π
Amplitude: |1| = 1
Reflection: No negative sign, so NO reflection
Phase Shift: 0/1 = 0, so NO phase shift
Vertical Shift: D = 3, so YES - shift up 3 units
Transformations: Starting from y = sin x, shift the entire graph up 3 units.
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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 factor of 2/3
2. y = 4 sin πx
- Period: 2
- Amplitude: 4
- Reflection: none
- Phase Shift: none
- Vertical Shift: none
- Transformations: Horizontal compression (period becomes 2), then vertical stretch by factor of 4
3. y = 2 sin(1/2)x
- Period: 4π
- Amplitude: 2
- Reflection: none
- Phase Shift: none
- Vertical Shift: none
- Transformations: Horizontal stretch (period becomes 4π), then vertical stretch by factor of 2
4. y = sin x + 3
- Period: 2π
- Amplitude: 1
- Reflection: none
- Phase Shift: none
- Vertical Shift: up 3
- Transformations: Vertical shift up 3 units
Parent Tip: Review the logic above to help your child master the concept of sine and cosine graphs worksheet.
