Trigonometric Graph Transformations

Master sine and cosine waves. Manipulate amplitude, frequency/period, horizontal phase shifts, and vertical shifts, and listen to the mathematical curves.

Interactive Sandbox

Wave Graph Sandbox

Toggle between Sine and Cosine waves. Adjust sliders to observe vertical stretch/reflection (Amplitude), horizontal frequency stretch (Period), horizontal displacement (Phase Shift), and vertical translation (Midline).

Equation Display

y = 1.00 · sin(1.00(x - 0.00)) + 0.00
Amplitude (A) 1.00
Frequency Coefficient (B) 1.00
Phase Shift (C) 0.00 rad
Vertical Shift (D) 0.00
Calculated Properties:
• Period (T) = 2π / |B| = 6.28 units
• Amplitude = |A| = 1.00 units
• Midline (Vertical Axis) = y = 0.00
Audio Integration

Audible Sound Waves

Sound is physically caused by repeating variations in air pressure. Sine functions map perfect audio tones. Activate the audio synthesizer to hear how pitch directly correlates with frequency, and loudness corresponds to amplitude!

Synthesizer Control


Audio Playback status:
Synthesizer Waveform Timbre:
Synth Audio Mapping:
• Plotted Wave Frequency B = 1.00
• Synthesized Tone Pitch = 220Hz × B = 220.00 Hz (Vibrations/sec)
• Plotted Wave Amplitude A = 1.00
• Output Synthesizer Volume = 15%

Connecting Geometry to Physics


Air particles oscillate back and forth in a pattern modeled by trigonometric waves.

  • High Frequency (B ↑) → Pitch increases. More waves fit in a second.
  • High Amplitude (A ↑) → Sound gets louder. Peaks carry more energy.
  • Waveform Timbre → Harmonics alter wave geometry from smooth sines to angular shapes.
Wave Physics

Wave Superposition & Beats

When two waves occupy the same space, they sum together algebraically: $y_{sum} = y_1 + y_2$. Set two waves of slightly different frequencies to see and hear **constructive/destructive interference (beats)**, or set simple integer ratios to create **harmonic chords**.

Interference Plotter

Wave 1 Options
Amplitude A₁ 1.0
Freq B₁ 1.00 (220 Hz)
Wave 2 Options
Amplitude A₂ 1.0
Freq B₂ 1.20 (264 Hz)
Try this! Set B₁ = 1.00 and B₂ = 1.05 to watch the combined wave form "amplitude packets" and hear a slow pulsing beat (interference envelope). Set B₁ = 1.0 and B₂ = 1.5 to hear a perfect musical fifth!
Self Evaluation

Conceptual Quiz

Assess your understanding of trigonometric transformations. Solve the questions below to test your wave mastery.

Question 1 of 5 Score: 0/0

For the wave equation y = 3 sin(2(x - π)) + 4, what is the amplitude of the function?

Correct Answer!

Indeed, the amplitude is dictated by the absolute value of coefficient A. Here A = 3, so amplitude is 3.

Trig Focus Areas


  • Period Calculation (2π/B)
  • Amplitude Boundaries
  • Phase Shift (Translations)
  • Wave Superposition Physics