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Math101Phase 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.
