Amplitude, period, phase shift, and midline
Amplitude, Period, Phase Shift, and Midline
Amplitude, Period, Phase Shift, and Midline is a topic in Trigonometric Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(G), P.2(O), which requires students to graph the transformations of trigonometric functions.
Amplitude, period, phase shift, and midline are the four parameters of a sinusoid \(y=a\sin(b(x-c))+d\): \(|a|\), \(\dfrac{2\pi}{b}\), \(c\), and \(y=d\).
Theory
A general sinusoid is
and each constant controls one feature:
- \(|a|\) — amplitude: half the distance from peak to trough.
- \(b\) — period \(=\dfrac{2\pi}{b}\): how long one cycle takes.
- \(c\) — phase shift: horizontal slide (right if \(c>0\)).
- \(d\) — midline \(y=d\): the horizontal center line the wave oscillates about.
From \(y=a\sin(b(x-c))+d\):
How to analyze or build a sinusoid
- Amplitude and midline: from \(|a|\) and \(d\), or from max/min.
- Period: compute \(\dfrac{2\pi}{b}\), or read the cycle length off a graph and solve for \(b\).
- Phase shift: factor the input as \(b(x-c)\) and read \(c\).
- Assemble \(y=a\sin(b(x-c))+d\) (or cosine).
Compare with \(y=a\sin(b(x-c))+d\).
| \(\text{amplitude}=|a|\) | \(=\) | \(3\) |
| \(\text{period}=\dfrac{2\pi}{b}\) | \(=\) | \(\dfrac{2\pi}{2}=\pi\) |
| \(\text{phase shift}=c\) | \(=\) | \(\dfrac{\pi}{4}\ \text{right}\) |
| \(\text{midline}=d\) | \(=\) | \(y=1\) |
Period \(=\dfrac{2\pi}{b}\) with \(b=\dfrac12\).
| \(\text{period}\) | \(=\) | \(\dfrac{2\pi}{1/2}=4\pi\) |
Amplitude is half the peak-to-trough distance; midline is their average.
| \(\text{amplitude}\) | \(=\) | \(\dfrac{9-1}{2}=4\) |
| \(\text{midline}\) | \(=\) | \(\dfrac{9+1}{2}=5\) |
Find \(b\) from the period: \(b=\dfrac{2\pi}{\text{period}}\).
| \(b\) | \(=\) | \(\dfrac{2\pi}{\pi}=2\) |
| \(y\) | \(=\) | \(2\cos(2x)-1\) |
Common pitfalls
Frequently asked questions
What do a, b, c, and d control in a sinusoid?
In \(y=a\sin(b(x-c))+d\): \(|a|\) is amplitude, \(\dfrac{2\pi}{b}\) is period, \(c\) is phase shift, and \(d\) is the midline.
How do you find the period of a sinusoid?
Divide \(2\pi\) by \(b\), the coefficient of \(x\). A bigger \(b\) means a shorter period.
How do you find amplitude and midline from a graph?
Amplitude is half the distance between the maximum and minimum; the midline is their average.
What is a phase shift?
A horizontal slide of the wave. Writing the input as \(b(x-c)\), the graph shifts right by \(c\) (left if \(c\) is negative).