Math101 · Independent Ontario learning library
Math101learn.math101.caPhase Shift
Phase shift is horizontal displacement of a periodic graph. In $y=a\sin(b(x-h))+k$ or cosine form, the phase shift is $h$: right if $h>0$, left if $h<0$.
Phase shift aligns periodic models with real starting times, seasons, waves, and rotations. Correct factoring is crucial in modelling and equation solving.
Intuition and core definition
Phase shift is horizontal displacement of a periodic graph. In $y=a\sin(b(x-h))+k$ or cosine form, the phase shift is $h$: right if $h>0$, left if $h<0$. The expression must be factored into $b(x-h)$ before reading $h$.
Notation, language, and conditions
For $a\sin(bx-c)+k$ with $b\ne0$, factor $b$: $b[x-c/b]$, so phase shift is $c/b$, not $c$. The period is $2\pi/|b|$ for sine or cosine. If $a=0$, the output is constant and no unique phase can be identified; otherwise shifts differing by an integer period produce the same graph.
Why this idea matters
Phase shift aligns a periodic model's landmarks with observed starting times without changing its amplitude, period, or midline.
A dependable method
- Identify the complete trig argument.
- Factor the coefficient of $x$ to obtain $b(x-h)$.
- Solve the argument’s central reference equation $b(x-h)=0$ for $h$.
- Compute period independently and place quarter-period landmarks from $h$.
- Check by substituting the shifted landmark into the original function.
Worked example
Representations and interpretation
A phase shift slides the entire wave horizontally without changing amplitude, period, or midline. Matching a key feature—such as a sine midline crossing or cosine maximum—between base and transformed graphs reveals the displacement.
Reasoning about variations
Because waves are periodic, cosine shifted right by one full period is identical to the unshifted graph. Also, a negative amplitude can be traded for a half-period shift, so algebraically different parameter sets may describe the same graph.
Common mistakes
How to check your work
- Substitute $x=h$ and confirm the internal argument is zero.
- Compare period and amplitude before and after a pure horizontal translation.
- Locate a second landmark one quarter-period away.
Practice
- Find the phase shift of $y=\sin(3x-\pi)$.
- Find the phase shift of $y=\cos[2(x+4)]$.
- Does a pure phase shift change amplitude?
Answers and brief solutions
Show answers
- $\frac\pi3$ right $3x-\pi=3(x-\pi/3)$.
- $4$ left $x+4=x-(-4)$.
- No It changes horizontal location only.
Synthesis and transfer
For a seasonal temperature curve whose maximum occurs later than the base cosine maximum, match those landmarks and factor the argument before reading the horizontal displacement.
A model $T(t)=12+8\cos[2\pi(t-7)/365]$ places its maximum at day $7$ relative to the chosen origin and repeats every $365$ days. The quantity $t-7$ shifts the base cosine right, while the factor $2\pi/365$ controls period; confusing those roles would predict the wrong season length. Adding or subtracting $365$ from the shift gives an equivalent graph because the wave is periodic. A negative amplitude could also be traded for a half-period shift, so fitted parameters are not always unique. Comparing a named landmark, such as the first maximum in the observation window, gives phase a clear contextual interpretation.
Related topics
Teaching and accessibility note
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Phase Shift.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the phase shift of $y=\sin(3x-\pi)$.
- $3x-\pi=3(x-\pi/3)$.
End of lesson
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