Amplitude Formula

Amplitude is the maximum vertical distance from the midline of a periodic function to a peak or trough.

The Formula

amplitude=ymax⁡−ymin⁡2\text{amplitude}=\frac{y_{\max}-y_{\min}}{2}

When to use: Amplitude is the maximum displacement from the middle of a wave — it is half the total height of a full oscillation from crest to trough.

Quick Example

f(x)=3sin⁡(x)f(x) = 3\sin(x) has amplitude 3: it oscillates between −3-3 and +3+3. Doubling amplitude doubles the peak height but does not change the period.

Notation

In y=Asin⁡(Bx+C)+Dy=A\sin(Bx+C)+D, amplitude is ∣A∣|A|.

What This Formula Means

Amplitude is the maximum vertical distance from the midline of a periodic function to a peak or trough.

Amplitude is the maximum displacement from the middle of a wave — it is half the total height of a full oscillation from crest to trough.

Formal View

Amplitude can be formalized with precise domain conditions and rule-based inference.

Worked Examples

Example 1

easy
Find the amplitude of f(x)=5sin⁡(x)f(x) = 5\sin(x).

Answer

Amplitude=5\text{Amplitude} = 5

First step

1
The general form of a sine function is f(x)=Asin⁡(Bx+C)+Df(x) = A\sin(Bx + C) + D, where ∣A∣|A| is the amplitude.

Full solution

  1. 2
    Here A=5A = 5, so the amplitude is ∣5∣=5|5| = 5.
  2. 3
    This means the graph oscillates between y=−5y = -5 and y=5y = 5.
Amplitude is the distance from the midline to the maximum (or minimum) of a periodic function. For y=Asin⁡(x)y = A\sin(x) or y=Acos⁡(x)y = A\cos(x), the amplitude is ∣A∣|A|. It measures how far the wave deviates from its center position.

Example 2

medium
Find the amplitude and midline of g(x)=−3cos⁡(2x)+4g(x) = -3\cos(2x) + 4.

Example 3

medium
Find the amplitude, period, and midline of y=2sin⁡(3x)−4y = 2\sin(3x) - 4.

Common Mistakes

  • Using the full crest-to-trough height as amplitude - amplitude is half that: ymax−ymin2\frac{y_{max}-y_{min}}{2}.
  • Taking AA as signed when it can be negative - amplitude is ∣A∣|A|, always nonnegative.
  • Confusing amplitude with period - amplitude is vertical (height), period is horizontal (cycle length).

Why This Formula Matters

It separates the strength of an oscillation (how loud a sound, how big a tide) from its timing — confusing it with period or frequency means misreading every sinusoidal model in physics, sound, and signal processing. Recognizing it by "Am I measuring the vertical distance from the midline to a peak (half the total swing)?" — rather than by familiar numbers — is what lets a student tell it apart from period and frequency and midline / vertical shift (dd) in a mixed problem set.

Frequently Asked Questions

What is the Amplitude formula?

Amplitude is the maximum vertical distance from the midline of a periodic function to a peak or trough.

How do you use the Amplitude formula?

Amplitude is the maximum displacement from the middle of a wave — it is half the total height of a full oscillation from crest to trough.

What do the symbols mean in the Amplitude formula?

In y=Asin⁡(Bx+C)+Dy=A\sin(Bx+C)+D, amplitude is ∣A∣|A|.

Why is the Amplitude formula important in Math?

It separates the strength of an oscillation (how loud a sound, how big a tide) from its timing — confusing it with period or frequency means misreading every sinusoidal model in physics, sound, and signal processing. Recognizing it by "Am I measuring the vertical distance from the midline to a peak (half the total swing)?" — rather than by familiar numbers — is what lets a student tell it apart from period and frequency and midline / vertical shift (dd) in a mixed problem set.

What do students get wrong about Amplitude?

The procedure for amplitude is the easy part; the trap is using the full crest-to-trough height as amplitude. Asking "Am I measuring the vertical distance from the midline to a peak (half the total swing)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Amplitude formula?

Before studying the Amplitude formula, you should understand: periodic functions, transformation, scaling functions.

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