Key takeaway: The period of a function is the smallest positive horizontal distance after which its graph repeats exactly, equal to 2π/|B| for y = sin(Bx) and y = cos(Bx).
Grade 11 – Algebra 2F-TF.B.5Trigonometry
Definition of Period of a Function
The period of a function is the length of one complete cycle of a repeating graph. Slide the graph to the right by exactly that distance and it lands on top of itself.
Many everyday patterns repeat: a Ferris wheel returns to the ground, a heartbeat returns to its start, and the seasons come around. Sine and cosine have a basic cycle length of \(2\pi\). A number multiplying the input squeezes or stretches the cycle: a bigger multiplier means faster repetition and a shorter cycle. Only the smallest repeating distance counts as the answer.
Formula and notation
A function \(f\) has period \(P\) when \(f(x+P)=f(x)\) for every \(x\). For the trig family:
\[\text{sine, cosine: } P=\frac{2\pi}{|B|}, \qquad \text{tangent: } P=\frac{\pi}{|B|}.\]
Here \(B\) is the number multiplying \(x\) inside the function.
Key facts
- The period of a function ignores amplitude, vertical shifts and reflections.
- Frequency, the number of cycles per unit, is the reciprocal of this length.
- Tangent repeats twice as fast as sine because its basic cycle is only π.
- Any multiple of a repeating distance also works, but only the smallest is reported.
Period of a Function example
Find the period of \(f(x)=5\sin(\frac{\pi x}{4})\) and of \(g(x)=\tan(3x)\).
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For \(f\), the multiplier is \(B=\frac{\pi}{4}\). Then \(P=2\pi: \frac{\pi}{4}=2\pi\cdot\frac{4}{\pi}=8\).
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For \(g\), the multiplier is \(B=3\) and tangent uses \(\pi\): \(P=\frac{\pi}{3}\).
\(f\) repeats every 8 units and \(g\) repeats every \(\frac{\pi}{3}\) units.
Common mistake
Watch out
Hugo claims that \(\sin(3x)\) has period 3, since the multiplier is 3.
Why it is wrong
The multiplier divides the basic cycle, it does not replace it, so the period of a function like this one is smaller than 2π. The cycle length is \(\frac{2\pi}{3}\approx 2.09\). Check: at \(x=0\) and \(x=\frac{2\pi}{3}\) the inside of the sine is 0 and \(2\pi\), the same output.
Related terms and lessons
- Algebra 2 · Graphs of Trigonometric FunctionsLessonWorksheetQuiz
- Precalculus · Graphs of Trigonometric FunctionsLessonWorksheetQuiz
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