Amplitude Formula

The Formula

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

Notation

In y=Asin(Bx+C)+D, amplitude is |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) = 5\sin(x).

Solution

  1. 1
    The general form of a sine function is f(x) = A\sin(Bx + C) + D, where |A| is the amplitude.
  2. 2
    Here A = 5, so the amplitude is |5| = 5.
  3. 3
    This means the graph oscillates between y = -5 and y = 5.

Answer

\text{Amplitude} = 5
Amplitude is the distance from the midline to the maximum (or minimum) of a periodic function. For y = A\sin(x) or y = A\cos(x), the amplitude is |A|. It measures how far the wave deviates from its center position.

Example 2

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

Common Mistakes

  • Reading amplitude as the coefficient including sign
  • Computing full peak-to-trough distance instead of half

Why This Formula Matters

Amplitude determines how large a wave is — in physics it corresponds to energy (sound loudness, wave intensity), and in modeling it sets the scale of oscillation.

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)+D, amplitude is |A|.

Why is the Amplitude formula important in Math?

Amplitude determines how large a wave is — in physics it corresponds to energy (sound loudness, wave intensity), and in modeling it sets the scale of oscillation.

What do students get wrong about Amplitude?

Amplitude is always non-negative — a negative coefficient like -3\sin(x) gives amplitude 3, not -3; the negative reflects the graph but the amplitude is |-3| = 3.

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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