Lesson 2: Phase Shift, Vertical Shift, and Transformations
Estimated time: 35-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Identify the phase shift C/B and vertical shift D in y = A sin(Bx − C) + D
- Graph sinusoidal functions with all four parameters (A, B, C, D)
- Determine the equation of a sinusoidal function from its graph
- Apply these concepts to model periodic real-world phenomena
The General Sinusoidal Form
y = A sin(Bx − C) + D or y = A cos(Bx − C) + D
|A| = amplitude | 2π/|B| = period | C/B = phase shift | D = vertical shift (midline)
Phase shift moves the graph left or right. If C/B > 0, the shift is to the right. If C/B < 0, the shift is to the left.
Vertical shift D moves the entire graph up (D > 0) or down (D < 0). The midline becomes y = D.
Phase Shift (Horizontal Translation)
Example 1: Phase Shift
Graph y = sin(x − π/4). Identify the phase shift.
Solution: B = 1, C = π/4. Phase shift = C/B = π/4 to the right.
The entire sine graph shifts π/4 units to the right. New key starting point: (π/4, 0) instead of (0, 0).
Example 2: Phase Shift with B ≠ 1
Find the phase shift of y = 3 cos(2x − π).
Solution: B = 2, C = π. Phase shift = π/2 to the right. Period = 2π/2 = π.
The starting interval shifts from [0, π] to [π/2, 3π/2].
Vertical Shift
Example 3: Vertical Shift
Graph y = sin x + 3.
Solution: The midline shifts from y = 0 to y = 3. The graph oscillates between 3 − 1 = 2 and 3 + 1 = 4.
Example 4: All Parameters
For y = 2 sin(3x − π) + 1, identify all parameters and key features.
Amplitude: |2| = 2 Period: 2π/3 Phase shift: π/3 right Vertical shift: 1 (midline y = 1)
Range: [1 − 2, 1 + 2] = [−1, 3]
Starting interval: [π/3, π/3 + 2π/3] = [π/3, π]
Graphing Strategy for the General Form
- Find amplitude |A|, period 2π/|B|, phase shift C/B, and vertical shift D
- Draw the midline y = D
- The graph starts at x = C/B and ends one period later at x = C/B + period
- Divide this interval into 4 equal parts to get 5 key x-values
- Plot the 5 key points using the sin or cos pattern, scaled by A and shifted by D
- Connect with a smooth curve and extend
Finding the Equation from a Graph
Example 5: Writing the Equation
A sinusoidal graph has midline y = 2, amplitude 3, period π, and starts at its maximum at x = π/4. Write the equation.
Solution: Since it starts at a max, use cosine. D = 2, A = 3, period = π so B = 2π/π = 2. Phase shift = π/4, so C = B × π/4 = π/2.
y = 3 cos(2x − π/2) + 2
Check Your Understanding
1. Find amplitude, period, phase shift, and vertical shift for y = −4 cos(2x + π) − 1.
2. What is the range of y = 5 sin(x) + 3?
3. A sine curve has period 4π and phase shift π to the right. What are B and C?
Key Takeaways
- Phase shift = C/B: positive shifts right, negative shifts left.
- Vertical shift D moves the midline to y = D.
- The range becomes [D − |A|, D + |A|].
- One period runs from x = C/B to x = C/B + 2π/|B|.