Periodic Functions: Period, Amplitude, and Axis

A function is periodic when there is a fixed horizontal distance $p$ (the period) such that shifting any point $p$ units to the right lands you on a point with exactly the same $y$-value, and this holds everywhere on the graph. One complete piece of the graph that then repeats is called a CYCLE. To measure the period, choose a point that is easy to recognize, such as a peak, and measure horizontally to the next point that is identical in EVERY way: the same height, and doing the same thing (rising, falling, or flat). Using the next peak works well because peaks are easy to spot. Measuring between two points at the same height can go wrong, because in one cycle the graph usually passes each height twice, once going up and once going down. In the figure, the graph reaches its first flat top at $x=2$ and the next at $x=8$, so the period is $6$. Graphs that never settle into repetition, like a population that keeps growing or a car's odometer reading, are not periodic even if they wiggle.

• Period: the horizontal length of ONE full cycle before the pattern starts repeating exactly. It carries the units of the horizontal axis (seconds, hours, days, degrees).
• Maximum and minimum: the highest and lowest output values the graph ever reaches.
• Axis (or midline): the horizontal line exactly halfway between the maximum and minimum — the graph oscillates evenly above and below it. Write it as an equation, such as $y=5$.
• Amplitude: the distance from the axis up to the maximum (equivalently, down to the minimum) — always a POSITIVE number, half the total vertical range.

Why these formulas? The axis sits exactly halfway between the top and the bottom, and the halfway point of two numbers is their average, so you add and divide by $2$. The total vertical range, $\max -\min$, runs from the bottom through the axis to the top, and the axis cuts it into two equal halves; the amplitude is one of those halves. In the figure, $\text{axis}=\frac{8+2}{2}=5$ and $\text{amplitude}=\frac{8-2}{2}=3$. Two consequences are handy for checking your work: $\max =\text{axis}+\text{amplitude}$ and $\min =\text{axis}-\text{amplitude}$. So if you know the axis and ONE extreme value, you can find the other extreme without seeing the whole graph. These four quantities can be read directly off a graph, or extracted from a table or description of a real situation, without ever writing down the equation of the curve.

Periodic Functions (full lesson) | math 11 | jensenmath.ca

Real data usually arrives as a table. To find the period from a table, look for the time between two successive maximums (or two successive minimums), not between a maximum and the next minimum, which is only half a cycle. The table below records the water depth at a dock every three hours: the depth peaks at $5$ m at hours $3$ and $15$, so the period is $15-3=12$ hours; the axis is $\frac{5+1}{2}=3$ m and the amplitude is $\frac{5-1}{2}=2$ m. Because the pattern repeats, the period also lets you predict far into the future. To find the depth at hour $51$, subtract whole periods: $51-4(12)=3$, so the depth at hour $51$ equals the depth at hour $3$, which is $5$ m. The same trick works on a graph: in the figure, $f(40)=f(40-6\times 6)=f(4)=2$.