Period of a Function
In short: The period of a function is the shortest stretch of the x-axis over which one full cycle plays out before the identical pattern starts again, written f(x + P) = f(x).
Algebra 2TrigonometryF-TF.B.5
What is Period of a Function?
The period of a function is the length of its repeating pattern. A Ferris wheel returns a rider to the same height after every full turn, and the time for one turn is the period of the rider’s height.
Formally, \(P\) works when \(f(x + P) = f(x)\) for all \(x\), and it must be the least positive number that does. For \(y = a\sin(bx)\) and \(y = a\cos(bx)\) it equals \(\frac{2\pi}{|b|}\); for \(y = a\tan(bx)\) it equals \(\frac{\pi}{|b|}\).
How to spot it
- A graph shows the same wave shape over and over, and you are asked for the period of a function, meaning how long one repeat is.
- A formula has a number multiplying x inside a sine, cosine or tangent, as in cos(πx/4).
- A story describes something that cycles, like tides, a pendulum or a rotating wheel.
- You are told that f(x + 15) = f(x) and asked to evaluate the function at a far-away input.
The method
- Rewrite the function so that the inside of the trig function is \(bx\) plus a shift.
- For sine and cosine compute \(\frac{2\pi}{|b|}\); for tangent compute \(\frac{\pi}{|b|}\).
- On a graph, measure the horizontal gap between two matching points, such as two consecutive peaks.
- To evaluate far away, subtract whole multiples of the period until the input lands in a known interval.
Period of a Function: solved example
Problem
Find the period of \(f(x) = \cos\left(\frac{\pi x}{4}\right)\) and of \(g(x) = \tan\left(\frac{2x}{3}\right)\).
Solution
For the cosine, \(b = \frac{\pi}{4}\), so the period is \(\frac{2\pi}{\pi/4} = 8\). For the tangent, \(b = \frac{2}{3}\), so the period is \(\frac{\pi}{2/3} = \frac{3\pi}{2}\).
The periods are 8 and \(\frac{3\pi}{2}\).
Try it
Question 1
Find the period of \(y = \sin(6x)\).
Show the answer
\(\frac{2\pi}{6} = \frac{\pi}{3}\).
Question 2
A function has period 15 and \(f(10) = 4\). Find \(f(55)\).
Show the answer
\(55 = 10 + 3(15)\), three full periods later, so \(f(55) = 4\).
Period of a Function vs. Amplitude
The period is a horizontal length: how far to travel along the x-axis before the pattern repeats. Amplitude is a vertical size: how far the curve moves from its midline. A wave can be tall but slow, or short but fast, because the two numbers are independent.
Keep going
Lessons
Worksheets and quizzes
- Graphs of Trigonometric Functions and Identities – Worksheet
- Graphs of Trigonometric Functions and Identities – Quiz
- Graphs of Trigonometric Functions – Worksheet
- Graphs of Trigonometric Functions – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Trigonometry