To find the amplitude of a wave or periodic function, you can use the following methods:
1. Using Maximum and Minimum Values
- Identify the maximum value (peak) and minimum value (trough) of the function.
- Calculate the amplitude as half the difference between the maximum and minimum values:
Amplitude=max value−min value2\text{Amplitude}=\frac{\text{max value}-\text{min value}}{2}Amplitude=2max value−min value
This formula gives the vertical distance from the midline (equilibrium position) to either the peak or the trough
2. Using the Midline
- Find the midline, which is the average of the maximum and minimum values:
Midline=max value+min value2\text{Midline}=\frac{\text{max value}+\text{min value}}{2}Midline=2max value+min value
- Then, calculate amplitude as the distance from the midline to the maximum value:
Amplitude=max value−midline\text{Amplitude}=\text{max value}-\text{midline}Amplitude=max value−midline
This method is equivalent to the first and useful when the function has a vertical offset
3. From the Equation of a Sine or Cosine Function
If the function is given in the form:
y=asin(b(x−h))+kory=acos(b(x−h))+ky=a\sin(b(x-h))+k\quad \text{or}\quad y=a\cos(b(x-h))+ky=asin(b(x−h))+kory=acos(b(x−h))+k
then the amplitude is the absolute value of the coefficient aaa:
Amplitude=∣a∣\text{Amplitude}=|a|Amplitude=∣a∣
This represents the vertical stretch or compression of the sine or cosine wave
Summary
- Amplitude is the height from the midline to a peak or trough.
- Formula using max and min values: max−min2\frac{\text{max}-\text{min}}{2}2max−min
- Formula using midline: max−midline\text{max}-\text{midline}max−midline
- From equation y=asin(…)+ky=a\sin(\ldots)+ky=asin(…)+k, amplitude = ∣a∣|a|∣a∣
These methods apply to sine, cosine, and other periodic functions or waves
