Key takeaway: The phase shift of a trig graph is its horizontal displacement from the basic wave, equal to C ÷ B for y = A sin(Bx − C) + D, to the right when positive.
Grade 12 – PrecalculusF-BF.B.3Trigonometry
Definition of Phase Shift
A phase shift slides a sine or cosine wave left or right without changing its height, its period or its center line. The wave looks the same but starts its cycle at a different horizontal position.
Spotting the shift takes one careful move. The equation must show the input as \(B(x-h)\), with \(B\) factored out of the bracket. Only then is the number \(h\) the true shift. If the bracket is left as \(Bx-C\), the number \(C\) is the shift measured in the scaled input, and you must divide by \(B\) to get the real horizontal distance.
Formula and notation
For \(y=A\sin(Bx-C)+D\) with \(B>0\):
\[\text{phase shift}=\frac{C}{B},\qquad y=A\sin\bigl(B(x-\tfrac{C}{B})\bigr)+D.\]
A positive result moves the graph right; a negative result moves it left. The period is \(\frac{2\pi}{B}\).
Key facts
- Factor B out of the bracket before reading the shift.
- A shift of one full period returns the graph to where it started.
- A cosine curve is a sine curve with a phase shift of π/2 to the left.
- The shift moves key points horizontally but leaves amplitude and midline unchanged.
Phase Shift example
Find the phase shift and the period of \(y=3\sin(2x-\frac{\pi}{3})+1\).
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Factor out the 2: \(2x-\frac{\pi}{3}=2(x-\frac{\pi}{6})\).
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The bracket is \(x-\frac{\pi}{6}\), so the shift is \(\frac{\pi}{6}\) to the right. Check: \(\frac{C}{B}=\frac{\pi/3}{2}=\frac{\pi}{6}\).
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The period is \(\frac{2\pi}{2}=\pi\).
The graph shifts \(\frac{\pi}{6}\) to the right and repeats every \(\pi\).
Common mistake
Watch out
Dimitri sees \(2x-\frac{\pi}{3}\) and states that the shift is \(\frac{\pi}{3}\) to the right.
Why it is wrong
The \(\frac{\pi}{3}\) is attached to the scaled input \(2x\), not to \(x\). Divide by 2 to get the horizontal shift: \(\frac{\pi}{6}\). Moving \(x\) by \(\frac{\pi}{3}\) would change the input by \(\frac{2\pi}{3}\) inside the sine.
Related terms and lessons
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