Trigonometric Graphs
Trigonometric graphs represent periodic functions that repeat their values in regular intervals, with sine and cosine functions producing characteristic wave patterns. The general form y = A sin(Bx + C) + D contains four parameters that control the wave's amplitude, period, horizontal shift, and vertical position. These graphs model cyclical phenomena ranging from sound waves to seasonal temperature variations.
Why it matters
Trigonometric graphs model countless real-world phenomena that exhibit periodic behavior. Sound engineers use sine waves to analyze audio frequencies, where a 440 Hz tone completes 440 cycles per second. Electrical engineers rely on these graphs to represent alternating current, which typically oscillates at 60 Hz in North America. Oceanographers use trigonometric functions to predict tide heights, with some locations experiencing tidal ranges exceeding 50 feet. Astronomers apply these concepts to model planetary orbits and stellar brightness variations. In physics, simple harmonic motion follows trigonometric patterns, from pendulum swings to spring oscillations. Climate scientists use periodic functions to analyze temperature cycles, seasonal variations, and long-term weather patterns that repeat over months or years.
How to solve trigonometric graphs
Trig Graphs — A sin(Bx + C) + D
- Amplitude = |A|. Vertical stretch/compression.
- Period = 2π/|B| (π/|B| for tan).
- Phase shift = −C/B (horizontal shift; + is left, − is right).
- Vertical shift = D; midline y = D; max = D + |A|, min = D − |A|.
Example: y = 2 sin(3x − π) + 1: amp=2, period=2π/3, shift=π/3 right, midline y=1.
Worked examples
What is the period of y = sin(3x)?
Answer: 2π/3
- Identify the period → period = 2π/3 — For sin(Bx), the period is 2π divided by the coefficient B. Here B = 3, so period = 2π/3 = 2π/3.
Find the amplitude and period of y = 4 sin(2x).
Answer: amplitude = 4, period = π
- Amplitude is the leading coefficient → amplitude = 4 — |A| in y = A sin(Bx) gives the amplitude. Here A = 4.
- Period is 2π divided by the coefficient of x → period = 2π/2 = π — For sin, one full cycle spans 2π when the argument increases by 2π. With B = 2, the argument reaches 2π when x reaches 2π/2.
Find the amplitude, period, and phase shift of y = 4 sin(3x + π).
Answer: amplitude = 4, period = 2π/3, phase shift = π/3 to the left
- Amplitude from the leading coefficient → amplitude = 4 — |A| = 4
- Period = 2π / |B| → period = 2π/3 — B = 3, so period = 2π/3 = 2π/3.
- Phase shift = −C / B → phase shift = π/3 to the left — The argument is B x + C with B = 3 and C = π. Phase shift is −C/B, which moves the graph horizontally. Positive shift = right; negative = left.
Common mistakes
- Confusing period calculation leads to writing period = 2B instead of period = 2π/B, so for y = sin(3x), incorrectly stating the period as 6 instead of 2π/3
- Phase shift direction errors result in claiming y = sin(x + π/2) shifts right by π/2 units instead of left by π/2 units, since the shift is -C/B = -π/2
- Amplitude confusion occurs when interpreting y = -3 sin(x) as having amplitude -3 instead of amplitude 3, since amplitude equals |A| and is always positive