Periodic Functions

Lots of things repeat: the height of a seat on a Ferris wheel, the tides, the phases of the moon, your heartbeat. A function whose graph repeats in a regular pattern is periodic. Once you know one cycle, you can predict the function’s value at any time, past or future.

Key ideas

Vocabulary

• A periodic function repeats its values at regular intervals.
• One complete repetition of the pattern is a cycle.
• The period is the horizontal length of one cycle, often a time.
• The axis of the curve is the horizontal line halfway between the maximum and minimum:

$$y = \frac{\text{max} + \text{min}}{2}$$

• The amplitude is the vertical distance from the axis to a maximum (or minimum):

$$\text{amplitude} = \frac{\text{max} - \text{min}}{2}$$

The amplitude is always positive.

A Ferris wheel rider’s height: period $10$ s, maximum $22$ m, minimum $2$ m.

Predicting with the period

Because the pattern repeats, adding or subtracting a whole number of periods doesn’t change the value:

$$f(x + \text{period}) = f(x)$$

To predict far into the future, subtract as many full periods as you can, then read the value from the first cycle.

Worked examples

Example 1: Reading a graph

Use the Ferris wheel graph above to find the period, maximum, minimum, axis, and amplitude.

Solution. The peaks are at $5$ s and $15$ s, so the period is $15 - 5 = 10$ s.

The maximum height is $22$ m and the minimum is $2$ m.

$$\text{axis: } h = \frac{22 + 2}{2} = 12, \qquad \text{amplitude} = \frac{22 - 2}{2} = 10$$

The axis is $h = 12$ m and the amplitude is $10$ m. (The wheel’s centre is $12$ m up, and its radius is $10$ m.)

Example 2: From a table

The water depth at a dock is measured every $2$ hours:

Time (h)	$0$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$
Depth (m)	$2.0$	$3.25$	$5.75$	$7.0$	$5.75$	$3.25$	$2.0$	$3.25$	$5.75$

Find the period, amplitude, and axis, and predict the depth at $30$ hours.

Solution. The depth is $2.0$ m at $0$ h and again at $12$ h, and the pattern repeats, so the period is $12$ h.

Maximum $7.0$, minimum $2.0$: amplitude $= \tfrac{7.0 - 2.0}{2} = 2.5$ m, axis $= \tfrac{7.0 + 2.0}{2} = 4.5$ m.

At $30$ h: subtract two periods, $30 - 24 = 6$, so the depth is the same as at $6$ h: $7.0$ m.

Example 3: Periodic or not?

Is each situation periodic?

(a) the height of the tip of a clock’s minute hand

(b) the height of a tree over many years

(c) the number of hours of daylight each day over several years

Solution.

(a) Periodic, with a period of $60$ minutes.

(b) Not periodic. A tree grows taller; it doesn’t repeat.

(c) Periodic, with a period of about $1$ year (about $365$ days).

Example 4: Extrapolating

A periodic function $f$ has a period of $8$, and $f(3) = 5$. Find $f(19)$ and $f(-5)$.

Solution. $19 = 3 + 2(8)$, so $f(19) = f(3) = 5$.

$-5 = 3 - 8$, so $f(-5) = f(3) = 5$.

Common mistakes

Measuring the period from a peak to a trough. That’s only half a cycle. Measure from one peak to the next peak, or between any two matching points.

Forgetting to halve. The amplitude is half the distance from maximum to minimum, not the whole distance.

Taking the axis as $0$. The axis is halfway between the maximum and minimum, which is often not the $x$-axis.

Predicting with a partial period. When extrapolating, subtract whole periods only.

Practice

1. (Warm-up) A periodic function has a maximum of $10$ and a minimum of $4$. Find its amplitude and the equation of its axis.

Solution

Amplitude $= \tfrac{10 - 4}{2} = 3$. Axis: $y = \tfrac{10 + 4}{2} = 7$.

2. (Warm-up) A resting heart beats once every $0.8$ seconds. What is the period of the heartbeat?

Solution

$0.8$ seconds.

3. (Warm-up) Which are periodic?

• (a) the height of a swinging pendulum’s bob
• (b) your height as you grow up
• (c) the phases of the moon

Solution

(a) and (c) are periodic. (b) isn’t: you don’t shrink and grow again in a cycle.

4. (Core) A periodic function has period $6$, and $f(2) = -1$. Find $f(14)$ and $f(-4)$.

Solution

$14 = 2 + 2(6)$ and $-4 = 2 - 6$, so both equal $f(2) = -1$.

5. (Core) A Ferris wheel has a diameter of $30$ m. Its lowest point is $2$ m above the ground, and it makes one revolution every $40$ s. Find the maximum height, amplitude, axis, and period of a rider’s height.

Solution

Maximum $= 2 + 30 = 32$ m, minimum $= 2$ m.

Amplitude $= \tfrac{32 - 2}{2} = 15$ m (the radius). Axis: $h = \tfrac{32 + 2}{2} = 17$ m (the height of the centre). Period: $40$ s.

6. (Core) Using the dock table in Example 2, predict the depth at $40$ hours.

Solution

$40 - 3(12) = 4$, so the depth matches the $4$-hour value: $5.75$ m.

7. (Core) A periodic function has a maximum of $3$ and an amplitude of $5$. What is its minimum, and what is its axis?

Solution

The axis is $5$ below the maximum: $y = 3 - 5 = -2$. The minimum is $5$ below the axis: $-2 - 5 = -7$.

8. (Challenge) A bike pedal is $17$ cm from the centre of the crank, and the centre is $30$ cm above the ground. The pedal makes one full turn every $0.75$ s. Find the maximum and minimum heights of the pedal, the amplitude, the axis, and the period, and describe the graph of its height over $3$ seconds.

Solution

Maximum $= 30 + 17 = 47$ cm, minimum $= 30 - 17 = 13$ cm. Amplitude $17$ cm, axis $h = 30$ cm, period $0.75$ s.

Over $3$ seconds the graph completes $\tfrac{3}{0.75} = 4$ full cycles, going between $13$ cm and $47$ cm.

9. (Challenge) Explain why the amplitude is half of (maximum − minimum), using the Ferris wheel as an example.

Solution

The axis is halfway between the maximum and minimum, so the distance from maximum to minimum is made of two equal parts: axis to maximum, and axis to minimum. The amplitude is one of those parts, so it’s half the total.

On a Ferris wheel, the maximum and minimum are a diameter apart, and the amplitude is the radius, half the diameter.