Amplitude
In short: Amplitude is the distance a sine or cosine wave rises above, or drops below, its midline, which equals half the gap between the maximum and the minimum, or |a| in y = a sin(bx).
Algebra 2TrigonometryF-TF.B.5
What is Amplitude?
The amplitude of a wave tells you how big the swing is. A calm lake and a stormy sea can have the same rhythm, but the stormy one has a far larger amplitude, because its crests rise much higher above the resting level.
For \(y = a\sin(bx + c) + d\) or \(y = a\cos(bx + c) + d\), the amplitude is \(|a|\). Equivalently, it is \(\frac{\text{maximum} – \text{minimum}}{2}\). It is never negative; a negative \(a\) only flips the wave upside down.
How to spot it
- A trig equation has a number multiplying the sine or cosine, such as -2.5 cos(4x).
- A graph is given and the question asks how far the peaks are from the middle of the wave.
- A story mentions high and low readings for tides, temperature or sound, and wants the size of the swing.
- You are asked to write an equation from a maximum value and a minimum value.
The method
- Rewrite the function so that it matches \(y = a\sin(\cdot) + d\) or \(y = a\cos(\cdot) + d\).
- Read the multiplier \(a\) and take its absolute value to get the swing.
- From a graph or data, compute \(\frac{\text{max} – \text{min}}{2}\) instead.
- Add and subtract the swing from the midline value to get the maximum and minimum.
Amplitude: solved example
Problem
For \(y = -2.5\cos(4x) + 7\), find the amplitude, the maximum value and the minimum value.
Solution
The multiplier is \(a = -2.5\), so the swing is \(|-2.5| = 2.5\). The vertical shift gives the midline \(y = 7\).
Maximum: \(7 + 2.5 = 9.5\). Minimum: \(7 – 2.5 = 4.5\). The amplitude is 2.5, with values ranging from 4.5 to 9.5.
Try it
Question 1
What is the amplitude of \(y = \frac{1}{2}\sin x\)?
Show the answer
The multiplier is \(\frac{1}{2}\), so the answer is \(\frac{1}{2}\).
Question 2
A sinusoid has maximum 8 and minimum -2. Find its amplitude and midline.
Show the answer
Swing: \(\frac{8 – (-2)}{2} = 5\). Midline: \(\frac{8 + (-2)}{2} = 3\), so \(y = 3\).
Amplitude vs. Period of a Function
This one measures vertical size: how far the wave climbs from the middle. The period measures horizontal length: how far along the x-axis one full cycle takes. Changing the number in front of sine alters the first and leaves the second alone, while changing the number multiplying x does the opposite.
Keep going
Lessons
Worksheets and quizzes
- Graphs of Trigonometric Functions and Identities – Worksheet
- Graphs of Trigonometric Functions and Identities – Quiz
- Graphs of Trigonometric Functions – Worksheet
- Graphs of Trigonometric Functions – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Trigonometry