Graphs of Polynominal Functions: Apply It

  • Graph polynomial functions.
  • Use the Intermediate Value Theorem.
  • Write the formula for a polynomial function given its graph.

Audio Waveform Analysis

Sound engineers and audio equipment designers use functions to model complex audio waveforms. By analyzing intercepts, graphing behavior, and using mathematical theorems, they can design better speakers, eliminate unwanted noise, and create high-quality audio equipment.

Audio waveforms cross zero displacement multiple times per cycle, and zero crossings (x-intercepts) are crucial for understanding sound quality and eliminating distortion.

Graphing Complete Waveform Behavior

Audio Engineering Insight: This waveform shows the string vibrates through 4 complete zero crossings in 4 milliseconds, indicating the fundamental frequency characteristics.

Writing Waveform Equations from Oscilloscope Data

An audio engineer observes a speaker waveform on an oscilloscope and needs to write its equation.

Use the observed data to write a polynomial model.

Observed data:

  • Zero crossings at [latex]t = -1, 2, 4[/latex]
  • At [latex]t = -1[/latex]: graph touches axis and bounces (doesn’t cross)
  • At [latex]t = 2[/latex] and [latex]t = 4[/latex]: graph crosses axis
  • Maximum amplitude: [latex]f(0) = 6[/latex]