How do Find Amplitude, Period, and Phase Shift?

Related Topics

• How to Find the Period of a Function?

Step-by-step guide to finding amplitude, period, and phase shift

Amplitude:

Amplitude refers to the maximum change of a variable from its mean value. The amplitude formula helps determine the \(sin\) and \(cos\) functions. Amplitude is represented by \(A\).

The sine function (or) the cosine function can be expressed as follows:

\(\color{blue}{\:x=A\:sin\:\left(ωt\:+\:ϕ\right)\:or\:x=A\:cos\:\left(ωt\:+\:ϕ\right)}\)

Here,

• \(x=\) displacement of wave (meter)
• \(A =\) amplitude
• \(ω =\) angular frequency (rad/s)
• \(t =\) time period
• \(ϕ =\) phase angle

The amplitude formula is also stated as the average of the \(max\) and \(min\) values of the \(sin\) or \(cos\) function. We always take the absolute value of the amplitude.

\(\color{blue}{Amplitude=\frac{max\:+\:min}{2}}\)

Phase shift:

Phase shift is the shift when the graph of the sine and cosine function moves from its usual position to the left or right, or we can say how much the function moves horizontally from its usual position in the phase shift.

The phase shift formula for a sinusoidal curve is shown below where the horizontal and vertical shifts are expressed. It can be positive or negative depending on the direction of change from the origin. The phase shift formula can be expressed as:

\(y\:=\:A\:sin\left(B\:\left(x\:+\:C\right)\right)+D\)
also, \(\left(F\left(x\right)=A\:sin\:\left(Bx\:−\:C\right)+D\right)\).

where

• \((\frac{C}{B}\)) shows the phase shift.
• \(A\) is the amplitude.

Period:

A function \(y = f(x)\) is called a periodic function if there exists a positive real number \(P\) such that \(f(x + P) = f(x)\), for all \(x\) belongs to the real numbers.

The periods of some important periodic functions are as follows:

• The period of \(sinx\) and \(cosx\) is \(2π\).
• The period of \(tanx\) and \(cotx\) is \(π\).
• The period of \(secx\) and \(cosecx\) is \(2π\).

Amplitude, Period, and Phase Shift – Example 1:

Find amplitude, period, and phase shift. \(3\:sin\left(4\left(x\:−\:0.5\right)\right)+5\)

Solution:

• amplitude \(A=3\)
• period \(\frac{2\pi \:}{B}=\frac{2\pi }{4}=\frac{\pi }{2}\)
• phase shift \(=−0.5\) (or \(0.5\) to the right)

Exercises for Amplitude, Period, and Phase Shift

Find the amplitude, period, and phase shift of each function.

1. \(\color{blue}{y=115\:sin\:\left(20\:\theta \right)+3}\)
2. \(\color{blue}{y=3\:cos\:\left(6x+\pi \right)}\)
3. \(\color{blue}{y=-4\:sin\:\left(\frac{2}{3}x-\frac{\pi }{3}\right)}\)

1. \(\color{blue}{amplitude =115, period =\frac{\pi}{10}}\)
2. \(\color{blue}{amplitude =3, period =\frac{\pi}{3}, phase\:shift=-\frac{\pi }{6}}\)
3. \(\color{blue}{amplitude =4, period =3\pi, phase\:shift=-\frac{\pi }{2}}\)