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Math101learn.math101.caAmplitude and Period
Amplitude measures a sinusoid's vertical departure from its midline, while period measures the input length of one complete 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
or cosine form, amplitude is
and the midline is
The maximum is $c+|a|$ and minimum is $c-|a|$.
Period
In degrees,
In radians,
The inside multiplier changes horizontal scale inversely: larger $|k|$ creates a shorter period and more frequent cycles.
Worked example
Quarter-period spacing is $30^\circ$.
From maximum and minimum
If a graph or dataset has maximum $M$ and minimum $m$,
Amplitude is half the total vertical range, not the maximum value.
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.
Frequency
Frequency counts cycles per input unit and is reciprocal to period:
If a machine completes $5$ cycles per second, period is $1/5$ second per cycle. Units reveal the reciprocal relationship.
Building a model
Identify midline and amplitude from extrema, period from repeated timing, and phase from a convenient starting landmark. Positive cosine is convenient at a maximum; positive sine is convenient at an upward midline crossing.
Equivalent sine and cosine equations can represent the same wave.
Context
Amplitude might represent half a temperature range, wheel radius, tide deviation, or signal strength. Period might be seconds, months, metres, or degrees depending on the input.
Interpret both with units and distinguish mathematical repetition from imperfect real cycles.
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?
Related topics
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Amplitude and Period.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For y = −4 cos(3(x − 20°)) + 2, what are the amplitude, period, and midline?
- Amplitude = |−4| = 4.
- Period = 360°/3 = 120°.
- The vertical shift gives midline y = 2.
End of lesson
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