Amplitude
Amplitude controls how “tall” the wave is. It's the first transformation parameter in the general sinusoidal equation — and the simplest to understand.
Introduction
The standard sine wave oscillates between and . But what if we need a wave that reaches or only ? The coefficient in stretches or compresses the wave vertically.
Past Knowledge
You graphed and with amplitude .
Today's Goal
Determine the amplitude from an equation and graph waves with different amplitudes.
Future Success
Amplitude is the first of four parameters in the general form .
Key Concepts
What is Amplitude?
For or :
Amplitude
The amplitude is always positive — it measures distance, not direction.
Visual Comparison
The Effect of
| Condition | Effect |
|---|---|
| Vertical stretch — wave gets taller | |
| Vertical compression — wave gets shorter | |
| Reflection over the -axis + stretch/compress |
Worked Examples
Identifying Amplitude
Question: What is the amplitude of ?
Step 1: Compare to . Here, .
Step 2: Amplitude .
Final Answer: Amplitude . The wave oscillates between and .
Negative Amplitude
Question: Describe the graph of .
Step 1: Amplitude .
Step 2: Since , the graph is reflected over the -axis.
Step 3: The wave starts at , goes down to first, then up to .
Final Answer: Amplitude , reflected. The wave oscillates between and but starts downward.
Finding the Equation from a Graph
Question: A cosine wave has a maximum of and a minimum of . Write its equation.
Step 1: Amplitude
Step 2: The midline is , so no vertical shift.
Final Answer:
Common Pitfalls
Saying the Amplitude is Negative
Amplitude is defined as — it is always positive. The amplitude of is , not . The negative sign means reflection, not negative amplitude.
Real-Life Applications
Ocean Tides
Tidal heights follow a sinusoidal pattern. A coastal area with high tides reaching feet and low tides at feet (relative to mean sea level) has a tidal amplitude of feet. Marine engineers use this to design docks, sea walls, and shipping lane depths.
Practice Quiz
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