Key takeaway: The amplitude of a periodic wave is half the distance between its highest and lowest values, which equals |A| in y = A sin(Bx) + D.
Grade 11 – Algebra 2F-TF.B.5Trigonometry
Definition of Amplitude
The amplitude of a wave is its strength: how far the graph climbs above, or drops below, the horizontal line running through its middle. A calm ripple has a small value; a storm swell has a large one.
For a sine or cosine function, the number in front of the trig part controls this height. Doubling that number doubles how far the wave stretches up and down. Reading it from a graph is easy: find the top of a crest, the bottom of a trough, and take half of the gap. The value is never negative, even when the wave is flipped upside down.
Formula and notation
For \(y=A\sin(Bx)+D\) or \(y=A\cos(Bx)+D\):
\[\text{amplitude}=|A|=\frac{y_{\max}-y_{\min}}{2}.\]
The constant \(D\) shifts the whole wave up or down but leaves the size unchanged, and \(B\) changes the horizontal length of a cycle.
Key facts
- The value is always positive; a negative A only reflects the wave over its midline.
- The maximum of the wave is the midline height plus A, the minimum is midline minus A.
- Changing B stretches the wave sideways without touching its height.
- Sound volume, tides and alternating voltage are all described by this value.
Amplitude example
A harbor tide gauge records a high of 9.4 feet and a low of 1.2 feet. Find the wave’s amplitude and its midline height.
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Subtract the extremes: \(9.4-1.2=8.2\).
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Halve the gap: \(\frac{8.2}{2}=4.1\).
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The middle height is the average of the extremes: \(\frac{9.4+1.2}{2}=5.3\).
The amplitude is 4.1 feet, around a midline at 5.3 feet.
Common mistake
Watch out
Sofia reads the same gauge and says the amplitude is the full gap, 8.2 feet.
Why it is wrong
The full gap is the peak-to-peak height. The amplitude is measured from the middle line to one extreme, so it is half of that: 4.1 feet. Check by adding it to the midline: \(5.3+4.1=9.4\).
Related terms and lessons
- Algebra 2 · Graphs of Trigonometric FunctionsLessonWorksheetQuiz
- Precalculus · Graphs of Trigonometric FunctionsLessonWorksheetQuiz
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