6.1Graphing Sine and Cosine Waves
Graph y = A·sin(Bx − C) + D and y = A·cos(Bx − C) + D. Identify amplitude |A|, period 2π/B, phase shift C/B, and vertical shift D. Write equations from graphs.
Sine and cosine waves model sound, light, tides, and electrical signals. Identifying amplitude, period, and phase shift from an equation or graph is a core skill for physics, engineering, and signal processing.
Essential Question
How do the parameters A, B, C, and D in y = A·sin(Bx − C) + D transform the basic sine graph?
Lesson Overview
The sine and cosine functions are the foundation of trigonometry. Their graphs are smooth, periodic waves called sinusoidal functions. By adjusting four parameters — A, B, C, and D — we can stretch, compress, shift, and reflect these waves to model real-world phenomena such as sound waves, tides, and electrical signals.
Amplitude
|A|
Height from midline to peak
Period
2π/B
Length of one full cycle
Phase Shift
C/B
Horizontal shift left/right
Vertical Shift
D
Midline y = D
The Basic Sine Function
y = sin(x) — amplitude 1, period 2π, starts at (0, 0)
The Basic Cosine Function
y = cos(x) — amplitude 1, period 2π, starts at (0, 1)
Effect of Amplitude (Parameter A)
Multiplying by A stretches or compresses the wave vertically. If A < 0, the graph reflects over the x-axis.
Effect of Phase Shift (Parameter C)
y = sin(x − π/2) shifts the sine wave π/2 units to the right. Note: y = sin(x − π/2) = cos(x).
Transformation Summary
Key Vocabulary
Sinusoidal function
A function of the form y = A·sin(Bx − C) + D or y = A·cos(Bx − C) + D. Its graph is a smooth, periodic wave.
Example: y = 3sin(2x − π) + 1 is sinusoidal
Amplitude
The maximum displacement from the midline. Amplitude = |A|. It is always positive.
Example: |A| = |−4| = 4 means the wave reaches 4 above and below the midline
Period
The horizontal length of one complete cycle of the wave. Period = 2π/B (for sine/cosine).
Example: If B = 2, period = 2π/2 = π
Phase shift
The horizontal translation of the graph. Phase shift = C/B. Positive means shift right; negative means shift left.
Example: y = sin(x − π/3): phase shift = (π/3)/1 = π/3 right
Vertical shift
The vertical translation of the graph. The midline moves from y = 0 to y = D.
Example: y = sin(x) + 2 has midline y = 2
Midline
The horizontal line y = D that runs through the middle of the sinusoidal wave, equidistant from the maximum and minimum.
Example: For y = 3sin(x) − 1, the midline is y = −1
Worked Examples
Identify A, B, C, D in y = 3sin(2x − π) + 1. Find the amplitude, period, phase shift, and vertical shift.
Write in standard form: y = A·sin(Bx − C) + D
Read off: A = 3, B = 2, C = π, D = 1
Amplitude = |A| = |3| = 3
Period = 2π/B = 2π/2 = π
Phase shift = C/B = π/2 (right)
Vertical shift = D = 1 (midline y = 1)
Graph y = 2cos(x). Plot the 5 key points and label the amplitude and period.
Identify: A = 2, B = 1, C = 0, D = 0
Amplitude = 2, Period = 2π
Key points for cos: (0,1), (π/2,0), (π,−1), (3π/2,0), (2π,1)
Multiply y-values by A = 2: (0,2), (π/2,0), (π,−2), (3π/2,0), (2π,2)
Plot these 5 points and connect with a smooth cosine curve
Graph y = sin(x + π/4). Identify the phase shift and plot the shifted key points.
Rewrite: y = sin(x − (−π/4)), so C = −π/4, B = 1
Phase shift = C/B = −π/4 (shift LEFT π/4 units)
Original key points: (0,0), (π/2,1), (π,0), (3π/2,−1), (2π,0)
Shift each x-value left by π/4: (−π/4, 0), (π/4, 1), (3π/4, 0), (5π/4, −1), (7π/4, 0)
Connect with a smooth sine curve
Write the equation of a sine function with amplitude 4, period π, phase shift π/3 right, and vertical shift −2.
Amplitude |A| = 4, so A = 4
Period = 2π/B = π → B = 2π/π = 2
Phase shift = C/B = π/3 → C = B · (π/3) = 2 · (π/3) = 2π/3
Vertical shift D = −2
Equation: y = 4·sin(2x − 2π/3) − 2
Graph y = −3sin(2x) + 1. Handle the reflection, amplitude, period, and vertical shift.
A = −3: amplitude = |−3| = 3; negative A means reflection over midline
B = 2: period = 2π/2 = π
C = 0, D = 1: no phase shift, midline y = 1
Key points for sin with period π: x = 0, π/4, π/2, 3π/4, π
y-values for sin: 0, 1, 0, −1, 0 → multiply by −3 and add 1: 1, −2, 1, 4, 1
Plot (0,1), (π/4,−2), (π/2,1), (3π/4,4), (π,1) and connect smoothly
Guided Practice
For y = 5sin(3x − π/2) − 4, identify A, B, C, D and find the amplitude, period, phase shift, and vertical shift.
Hint: Read A, B, C, D directly from the equation. Then apply: amplitude = |A|, period = 2π/B, phase shift = C/B, vertical shift = D.
Graph y = 3cos(2x). List the 5 key points after applying the amplitude and period.
Hint: Start with the basic cosine key points at x = 0, π/2, π, 3π/2, 2π. With B = 2, the period is π, so divide each x by 2. Then multiply y by 3.
Determine the phase shift of y = cos(x − π/6) and list the shifted key points.
Hint: Phase shift = C/B = (π/6)/1 = π/6 right. Add π/6 to each x-coordinate of the standard cosine key points.
Write the equation of a cosine function with amplitude 2, period 4π, phase shift π/2 left, and vertical shift 3.
Hint: Find B from period = 2π/B. Phase shift left means C/B is negative, so C is negative. Then assemble y = A·cos(Bx − C) + D.
Graph y = −2cos(x) + 3. Identify the reflection, midline, and maximum/minimum values.
Hint: A = −2 means reflection over the midline y = 3. Maximum = D + |A| = 3 + 2 = 5; minimum = D − |A| = 3 − 2 = 1.
Quick Check
Interactive Practice — 5 Questions
What is the amplitude of y = −4sin(3x) + 2?
What is the period of y = cos(πx)?
For y = sin(x − π/3), the phase shift is:
The midline of y = 2sin(x) − 5 is:
A sine function has amplitude 3, period 2π, no phase shift, and vertical shift 1. Its equation is:
Common Mistakes
Writing the period as B instead of 2π/B (e.g., saying period = 3 for y = sin(3x)).
Period = 2π/B. For y = sin(3x), period = 2π/3.
Confusing phase shift direction: y = sin(x − π/4) shifts LEFT.
y = sin(x − C/B) shifts RIGHT by C/B. Subtracting shifts right; adding shifts left.
Saying amplitude = A (including the sign), e.g., amplitude = −3 for y = −3sin(x).
Amplitude = |A|, always positive. For y = −3sin(x), amplitude = 3.
Forgetting to divide C by B when finding phase shift (e.g., phase shift = C instead of C/B).
Phase shift = C/B. For y = sin(2x − π), phase shift = π/2, not π.
Math Tips
Memorize the 5 key points for sin: (0,0),(π/2,1),(π,0),(3π/2,−1),(2π,0) and for cos: (0,1),(π/2,0),(π,−1),(3π/2,0),(2π,1).
To graph any sinusoidal function, find the 5 key x-values by dividing the period into 4 equal parts starting from the phase shift.
cos(x) = sin(x + π/2): cosine is just sine shifted left by π/2. You can always convert between them.
Maximum value = D + |A|; minimum value = D − |A|. These are quick checks for your graph.
If B is negative, use the identity sin(−θ) = −sin(θ) to rewrite with a positive B before identifying parameters.