Phase Shift

Sommaire

In short: A phase shift measures how far a sine or cosine wave has been pushed sideways along the x-axis; for a sine written with bx – c inside, the push is c divided by b.

PrecalculusTrigonometryF-BF.B.3

What is Phase Shift?

A phase shift tells you how far a wave has been pushed sideways. Two identical waves with different phases look the same except that one starts earlier along the x-axis.

Take \(y = a\sin(bx – c) + d\) or the cosine version. Factor the \(b\) out of the parentheses: \(y = a\sin\big(b(x – \frac{c}{b})\big) + d\). The amount \(\frac{c}{b}\) is the sideways slide, rightward when positive and leftward when negative. Skipping the factoring step is the classic slip, since \(bx – c\) moves the wave by \(\frac{c}{b}\) and not by \(c\).

How to spot it

A trig function has a number added to or subtracted from the inside, as in cos(2x – π/3).
A graph looks like a standard sine or cosine wave that starts later or earlier than usual.
A story says a cycle begins at a certain time offset, such as tides peaking at 2:30 a.m.
The question asks where the first maximum occurs, and the answer is not at x = 0.

The method

Write the function as \(a\sin(bx – c) + d\) or \(a\cos(bx – c) + d\).
Factor \(b\) out of the argument to reach the form \(b(x – h)\).
Read \(h = \frac{c}{b}\): positive means right, negative means left.
Slide the key points of the unshifted wave by \(h\) and redraw.

Because cos(2x – π/3) = cos(2(x – π/6)), the wave slides π/6 to the right.

Phase Shift: solved example

Problem

Find the phase shift and period of \(y = 4\cos\left(2x – \frac{\pi}{3}\right)\), and give the location of the first maximum for \(x \ge 0\).

Solution

Factor out the 2: \(y = 4\cos\left(2\left(x – \frac{\pi}{6}\right)\right)\). The slide is \(\frac{\pi}{6}\) to the right, and the period is \(\frac{2\pi}{2} = \pi\).

An unshifted cosine peaks at \(x = 0\), so this wave first peaks at \(x = \frac{\pi}{6}\). The answer is a phase shift of \(\frac{\pi}{6}\) right, with period \(\pi\).

Try it

Question 1

Describe the shift of \(y = \cos\left(x + \frac{\pi}{4}\right)\).

Show the answer

Here \(x + \frac{\pi}{4} = x – \left(-\frac{\pi}{4}\right)\), so the graph moves \(\frac{\pi}{4}\) to the left.

Question 2

Find the phase shift of \(y = 5\sin(3x – 9)\).

Show the answer

Factor: \(3(x – 3)\). The shift is 3 units to the right.

Phase Shift vs. Transformations of Functions

Sliding sideways is only one of several graph changes. The wider family also includes vertical shifts, stretches and reflections. The special term here is used for periodic waves, where the slide can be read as how far the cycle’s start has moved from where an ordinary sine or cosine begins.

Transformations of Functions

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