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Pre-Calculus Trigonometric functions

Amplitude, period, phase shift, and midline

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Amplitude, Period, Phase Shift, and Midline

California Pre-Calculus • Standard F-TF.5 • Trigonometric Functions

Amplitude, Period, Phase Shift, and Midline is a topic in Trigonometric Functions in the California Common Core State Standards. It is aligned to Standard F-TF.5, which requires students to interpret the amplitude, period, midline, and phase shift of a sinusoid.

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 \(y=d\).

California Pre-Calculus › Trigonometric Functions › Amplitude, Period, Phase Shift, and Midline  —  Standard F-TF.5

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Theory

A general sinusoid is

\[y=a\sin\big(b(x-c)\big)+d\quad\text{or}\quad y=a\cos\big(b(x-c)\big)+d,\]

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.
Factor out \(b\) first. The phase shift is \(c\) only when the input is written as \(b(x-c)\); otherwise divide.
Amplitude and midline of a sinusoid The graph of y equals 2 sine x plus 1 has amplitude 2 measured from the midline y equals 1 to a peak. x y midline y=1 amp 2
\(y=2\sin x+1\): amplitude 2 measured from the midline \(y=1\).
Period from the coefficient b The graph of y equals sine 2x completes two cycles between 0 and 2 pi, so its period is pi, half that of sine x. x y sin 2x sin x
\(y=\sin 2x\) has period \(\pi\) — two cycles where sine has one.

From \(y=a\sin(b(x-c))+d\):

\[\text{amplitude}=|a|,\quad \text{period}=\dfrac{2\pi}{b},\quad \text{phase shift}=c,\quad \text{midline}=y=d\]
amplitude is absolute value of a; period is 2 pi over b; phase shift is c; midline is y equals d
From max and min: amplitude \(=\dfrac{\max-\min}{2}\), midline \(=\dfrac{\max+\min}{2}\).

How to analyze or build a sinusoid

  1. Amplitude and midline: from \(|a|\) and \(d\), or from max/min.
  2. Period: compute \(\dfrac{2\pi}{b}\), or read the cycle length off a graph and solve for \(b\).
  3. Phase shift: factor the input as \(b(x-c)\) and read \(c\).
  4. Assemble \(y=a\sin(b(x-c))+d\) (or cosine).
Example 1 — Read off the parameters
For \(y=3\sin\!\big(2(x-\dfrac{\pi}{4})\big)+1\), give amplitude, period, phase shift, and midline.
Solution

Compare with \(y=a\sin(b(x-c))+d\).

\(\text{amplitude}=|a|\)\(=\)\(3\)
\(\text{period}=\dfrac{2\pi}{b}\)\(=\)\(\dfrac{2\pi}{2}=\pi\)
\(\text{phase shift}=c\)\(=\)\(\dfrac{\pi}{4}\ \text{right}\)
\(\text{midline}=d\)\(=\)\(y=1\)
amplitude 3, period pi, phase shift pi over 4 right, midline y equals 1
Example 2 — Period from b
Find the period of \(y=\cos\!\big(\dfrac{1}{2}x\big)\).
Solution

Period \(=\dfrac{2\pi}{b}\) with \(b=\dfrac12\).

\(\text{period}\)\(=\)\(\dfrac{2\pi}{1/2}=4\pi\)
period is 4 pi
Example 3 — Amplitude and midline from max/min
A sinusoid has maximum \(9\) and minimum \(1\). Find its amplitude and midline.
Solution

Amplitude is half the peak-to-trough distance; midline is their average.

\(\text{amplitude}\)\(=\)\(\dfrac{9-1}{2}=4\)
\(\text{midline}\)\(=\)\(\dfrac{9+1}{2}=5\)
amplitude 4, midline y equals 5
Example 4 — Write an equation
Write a cosine equation with amplitude 2, period \(\pi\), and midline \(y=-1\).
Solution

Find \(b\) from the period: \(b=\dfrac{2\pi}{\text{period}}\).

\(b\)\(=\)\(\dfrac{2\pi}{\pi}=2\)
\(y\)\(=\)\(2\cos(2x)-1\)
equation is y equals 2 cosine 2x minus 1

Common pitfalls

Factor out \(b\) before reading the phase shift. In \(\sin(2x-\pi)\) the shift is \(\dfrac{\pi}{2}\), not \(\pi\).
Period is \(\dfrac{2\pi}{b}\), not \(2\pi b\). A larger \(b\) gives a shorter period.
Amplitude is always positive. A negative \(a\) reflects the wave but the amplitude is \(|a|\).

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).

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