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Amplitude in Mathematics

Understanding the Height and Strength of Periodic Functions

In mathematics, specifically in trigonometry, Amplitude is the measure of the magnitude of a periodic function (like a wave). It represents the maximum displacement of points on a wave from its center line or equilibrium position.

[Image of amplitude of sine wave graph]

In simpler terms, amplitude tells us "how tall" the wave is. It is widely used in physics to describe the loudness of sound (higher amplitude = louder volume) or the brightness of light.

1. The Definition

Amplitude is strictly defined as the distance from the Midline (the horizontal center of the wave) to the Peak (maximum value) or to the Trough (minimum value).

Crucial Rule: Amplitude is always a positive number (distance), even if the graph goes downward.

2. Finding Amplitude from an Equation

For a standard trigonometric function in the form:

y = A sin(B(x - C)) + D
OR
y = A cos(B(x - C)) + D

The Amplitude is the absolute value of the coefficient A.

Amplitude = |A|

Examples:

  • For y = 3 sin(x), the amplitude is 3.
  • For y = -5 cos(x), the amplitude is 5 (not -5, because distance is positive).
  • For y = sin(x), the amplitude is 1 (since there is an invisible 1 in front).

3. Finding Amplitude from a Graph

If you are given a graph instead of an equation, you can calculate the amplitude by finding the highest point (Max) and the lowest point (Min).

[Image of amplitude formula max min]
Amplitude = (Max - Min) / 2

Example Calculation:

Imagine a wave that reaches a maximum height of 10 and a minimum height of -2.

  1. Max = 10, Min = -2.
  2. Difference = 10 - (-2) = 12.
  3. Divide by 2: 12 / 2 = 6.

The amplitude is 6. The "total height" of the wave (from bottom to top) is 12, but the amplitude is only the distance from the middle.

4. Real-World Applications

  • Sound: The amplitude of a sound wave determines its volume. A high amplitude means a loud sound; a low amplitude means a quiet sound.
  • Ocean Tides: The amplitude of a tide is the difference in water level between high tide and the average sea level.
  • Electricity: In alternating current (AC), the amplitude represents the peak voltage.

Conclusion

Understanding Amplitude is the first step in mastering periodic graphs. It essentially measures the "strength" or "intensity" of the function, telling us how far the graph stretches vertically from its resting position.