How do Find Amplitude, Period, and Phase Shift?

You can determine the amplitude, period, and phase shift of trigonometric functions easily! In this post, you will learn about this topic.

Tutor-style math help

How do Find Amplitude, Period, and Phase Shift: what to notice and how to work it

Trigonometry skill

A sine graph is a repeating wave. To sketch it well, mark the midline, amplitude, period, and five key points in one cycle.

What to notice first

For \(y=A\sin(Bx)+D\), the amplitude is \(|A|\), the period is \(2\pi/|B|\), and the midline is \(y=D\).

Common student mistake

Do not space the five key points randomly. Divide one period into four equal parts so the wave starts, rises, returns, falls, and returns again.

Key formulas and cues

\(y=A\sin(Bx)+D\)

\(\text{amplitude}=|A|\)

\(\text{period}=\frac{2\pi}{|B|}\)

\(\text{midline}=y=D\)

A reliable path

Choose the modelUse a right triangle, the unit circle, or a transformed graph.
Track unitsConvert degrees and radians when needed.
Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Read a sine graph rule

Example: \(y=3\sin(2x)-1\)

Amplitude is |3|.
Period is 2pi/2 = pi.
Midline is y = -1.

Answer: Amplitude \(3\), period \(\pi\), midline \(y=-1\).

Place five key points

Example: Graph one cycle of \(y=\sin x\).

Start at (0, 0).
Use quarter-period steps: pi/2, pi, 3pi/2, 2pi.
The y-values are 0, 1, 0, -1, 0.

Answer: \((0,0),(\pi/2,1),(\pi,0),(3\pi/2,-1),(2\pi,0)\).

Open pop-up practice

Try one before moving on

Try: Find the amplitude and period of \(y=2\sin(4x)\).

Answer: Amplitude \(2\), period \(\frac{\pi}{2}\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

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.

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

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

Effortless Math Team

3 weeks ago

Effortless Math Team

Related to This Article

Amplitude amplitude of function Period Period of a Function Phase Shift phase shift of a function

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