Amplitude and Period

Amplitude measures a sinusoid's vertical departure from its midline, while period measures the input length of one complete cycle.

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Amplitude, period, and phase shift describe a repeating graph. A wave repeats across the axis. A vertical measure marks amplitude, a horizontal bracket marks one period, and a second faint wave shows how a phase shift moves the cycle.

Amplitude answers “how far from the centre?” and period answers “how long until the pattern repeats?”

Parent functions

The parent functions $y=\sin x$ and $y=\cos x$ have amplitude $1$, midline $y=0$, maximum $1$, minimum $-1$, and period $360^\circ$ or $2\pi$ radians.

Their outputs repeat after each full period.

Transformed form

For

y=a\sin(k(x-d))+c

or cosine form, amplitude is

$$ |a|, $$

and the midline is

$$ y=c. $$

The maximum is $c+|a|$ and minimum is $c-|a|$.

Worked example

Quarter-period spacing is $30^\circ$.

Finding period from a graph

Measure horizontal distance between matching consecutive points: maximum to next maximum, minimum to next minimum, or upward midline crossing to next upward midline crossing.

Maximum to minimum is half a period, not a full period.

Common mistakes

Calling the maximum the amplitude. Subtract the midline or use half the range.

Calling $k$ the period. Period is inverse to $|k|$.

Measuring maximum to minimum as a full cycle. It is half.

Ignoring a negative $a$. It reflects the starting direction, not amplitude sign.

Mixing degree and radian period formulas. Match the angle unit.

Quick self-check

What are maximum, minimum, and midline?
Is amplitude half the vertical range?
What horizontal distance completes one full repeat?
Does $T=360^\circ/|k|$ or $2\pi/|k|$ match the unit?
Is the phase/start direction consistent with the formula?
Are amplitude and period interpreted with contextual units?