Tags: periodic function, period, amplitude, midline, increasing, decreasing, concave up, concave down, relative maximum, relative minimum, graph analysis, AP Precalculus Unit 3
A periodic function repeats its values at regular intervals. The length of one full cycle is the period. You can use the period to evaluate the function at very large or negative inputs by "wrapping" back into the visible window. Graph features like slope, concavity, and extrema also repeat every period.
Periodic function
A function f(x) where f(x) = f(x + p) for all x, with p being a positive constant. The output values repeat in a predictable cycle.
Period
The smallest positive value p such that f(x + p) = f(x) for every x. Visually, it is the horizontal length of one complete cycle of the graph.
Midline
The horizontal line halfway between the maximum and minimum values of a periodic function. Calculated as y = (max + min) / 2.
Amplitude
The vertical distance from the midline to the maximum (or minimum). Calculated as A = (max − min) / 2.
Relative maximum (local max)
A point where the function changes from increasing to decreasing. On the graph, a peak or hilltop.
Relative minimum (local min)
A point where the function changes from decreasing to increasing. On the graph, a valley or trough.
Concave up
The graph curves upward, like the inside of a bowl. The rate of change is increasing.
Concave down
The graph curves downward, like the top of a hill. The rate of change is decreasing.
Identify two consecutive points where the pattern begins to repeat (e.g. two consecutive peaks, two consecutive troughs, or two consecutive midline crossings in the same direction).
The horizontal distance between those two points is the period.
If the graph shows two complete periods from x = a to x = b, the period is (b − a) / 2.
Because f(x) = f(x + p) for all x, you can reduce any input to one that falls within the graphed window:
Divide the target x-value by the period.
Use the remainder to find the equivalent position within one cycle.
Read the function value from the graph at that equivalent position.
Example: if the period is 10 and you need f(34), compute 34 mod 10 = 4, so f(34) = f(4).
Positive slope at a point means the function is increasing there (graph rises left to right).
Negative slope means the function is decreasing (graph falls left to right).
Undefined slope occurs at sharp corners or cusps.
These behaviours repeat every period. To determine the slope at x = 102 when the period is 10, find 102 mod 10 = 2, then check the slope at x = 2 on the visible graph.
If an interval spans less than one period, determine which part of the cycle it maps to.
Use the remainder approach: for 64 < x < 65 with period 2.5, find 64 mod 2.5 = 1.5 and 65 mod 2.5 = 0, so the interval wraps around. Check the corresponding portion of the graph for curvature.
If the interval maps to a section that is entirely bowl-shaped, the function is concave up there. If hill-shaped, concave down. If it crosses an inflection point, both.
A relative max or min repeats every period.
To check whether x = 46 is an extremum when the period is 2.5, compute 46 mod 2.5 and locate that position on the graph. If it lands on a peak, it is a relative max. If on a valley, a relative min.
When given a table of values for a periodic function:
Look for the smallest interval after which the pattern of outputs repeats.
If f obtains a minimum at x = 2 and the next minimum is at x = 10, the period is 10 − 2 = 8.
The equation f(x) = f(x + p) then holds for all x, where p = 8.
Period from consecutive matching features: p = x₂ − x₁, where x₁ and x₂ are the x-values of two consecutive identical features (e.g. two peaks).
Reducing large inputs: f(x) = f(x mod p), provided you align the remainder with the graphed domain.
Midline: y = (max + min) / 2
Amplitude: A = (max − min) / 2
⚠️ The period is the distance between two identical points in the cycle (peak to peak, trough to trough), not peak to trough. Peak to trough is only half a period.
⚠️ When reducing large x-values, make sure the remainder maps correctly onto the graphed interval. If the graph starts at x = −5, you may need to shift accordingly rather than blindly using mod.
⚠️ "Increasing, decreasing, or both" questions test whether the interval spans a turning point. Map the interval back into one cycle before answering.
⚠️ "Concave up, concave down, or both" works the same way. The interval may cross an inflection point when mapped back.
⚠️ When identifying the period from a table, confirm the pattern by checking multiple values, not just the endpoints.
Q: A periodic function has a minimum at x = 2 and the next minimum at x = 10. What is the period?
A: The period is 10 − 2 = 8.
Q: If f has period 8 and f(2) = −3, what is f(26)?
A: 26 − 2 = 24, and 24 / 8 = 3 (a whole number of periods). So f(26) = f(2) = −3.
Q: A periodic function satisfies f(x) = f(x + p). If the period is 8, which equation is true: f(x) = f(x + 4) or f(x) = f(x + 8)?
A: f(x) = f(x + 8). The equation f(x) = f(x + 4) would mean the period is 4 (or a divisor of 4), which contradicts a period of 8.
Q: The graph of a periodic function shows two complete periods from x = −6 to x = 14. What is the period?
A: The total span is 14 − (−6) = 20. Two periods fit in that span, so the period is 20 / 2 = 10.
Q: On the interval 30 < x < 31, is a function with period 10 increasing, decreasing, or both, if the graph shows it is increasing at x = 0 to x = 1?
A: 30 mod 10 = 0 and 31 mod 10 = 1, so the interval 30 < x < 31 corresponds to 0 < x < 1. If the function is increasing on 0 < x < 1, it is also increasing on 30 < x < 31.
periodic function, period, cycle, repeating function, amplitude, midline, increasing function, decreasing function, concave up, concave down, relative max, relative min, local extrema, graph analysis, precalculus unit 3, AP Precalculus, Topics 3.1, 3.2, periodic behaviour, modular arithmetic for periodic functions