Key Definitions
The vocab you MUST know
Periodic Relationship
Output values demonstrate a repeating pattern over successive equal-length intervals of input values.
Period (p)
The length of x-values it takes for the function to complete one full cycle. The smallest such repeating interval.
f(x) = f(x − p)
A periodic function is a horizontal translation of itself. Shifting by one period gives the exact same graph.
e.g. if period = 5 → h(x) = h(x − 5)
Midline
The horizontal line halfway between the maximum and minimum output values. Center of the wave.
Midline = (max + min) / 2
Amplitude
Distance from the midline to the maximum (or minimum) value.
Amplitude = (max − min) / 2
Least Possible Period
The smallest valid repeating interval visible in the graph — look for where the pattern first starts over.
The Core Formula
How to reduce any input to a known value
Periodicity Reduction Formula
For a function with period p, you can add or subtract any multiple of p without changing the output:
f(a + n·p) = f(a) for any integer n
Strategy: Find how many periods fit, strip them away, and evaluate at the leftover value that's inside your known range.
All Inputs That Give a Specific Output
If f(a) = k for some value a, then every input x = a + n·p (where n is any integer) also gives output k.
This means there are infinitely many solutions, spaced exactly one period apart.
Worked Examples
Step by step from the notes
📌 Example 3a — g(14), period = 5. Known values include g(4) = 7.
1
Write: g(14) = g(4 + 2·5) — we added two full periods.2
Since adding multiples of the period doesn't change the output: g(4 + 10) = g(4).3
Look up g(4) from the table.g(14) = g(4) = 7 ✓
📌 Example 3c — g(−17), period = 5. Known values include g(3) = 7.
1
We need to get from −17 to a known input. −17 + 4·5 = −17 + 20 = 3 ✓2
g(−17) = g(3 − 4·5) = g(3). Adding 4 periods doesn't change the output.g(−17) = g(3) = 7 ✓
📌 Example 4d — h(5k − 3) where k is any integer. Period = 5, known h(7) = 2.
1
Rewrite: h(5k − 3) = h(−3 + 5k). The "5k" part is exactly k full periods!2
Strip the periods: h(−3 + 5k) = h(−3).3
Now reduce h(−3): −3 + 2·5 = 7, so h(−3) = h(7 − 2·5) = h(7).h(5k − 3) = h(7) = 2 for any integer k ✓
📌 Example 5 — Find ALL x where f(x) = 1. Period = 4, and f(2) = 1.
1
Identify one solution: f(2) = 1.2
Every integer multiple of the period away from x = 2 also works.All solutions: x = 2 + 4k, where k is any integer ✓
Graph Anatomy
What to look for on a periodic graph
📈 Annotated Periodic Function (Interactive)
Read the Period
Find a peak (or trough or crossing). Find the next identical point. The horizontal distance = period.
Concave Up ↑
Graph curves like a bowl. Rate of change is increasing on this interval.
Concave Down ↓
Graph curves like a hill. Rate of change is decreasing on this interval.
Positive vs Negative
Positive: output value is above the x-axis (y > 0). Negative: output value is below the x-axis (y < 0).
AP FRQ: Clock Problem (Example 6)
The FRQ 3 task model — know this cold
Setup
Clock with an 8-inch minute hand. Center is 120 inches above the floor. Clock runs 2× speed, so it completes a full revolution in 30 minutes instead of 60.
Period
30 minutes
Midline (center height)
120 inches
Maximum (hand pointing ↑)
120 + 8 = 128 in
Minimum (hand pointing ↓)
120 − 8 = 112 in
Amplitude
8 inches
F, G, J, K, P
(0,128) (7.5,120) (15,112) (22.5,120) (30,128)
FRQ Part (j) Answer: h is positive and decreasing on (t₁, t₂)
Between G and J, h falls from 120 to 112 — always above the x-axis (positive), and falling in value (decreasing). Answer: b.
FRQ Part (ii): Rate of change is increasing on (t₁, t₂)
The graph is concave up between G and J (the bottom of the curve), so the rate of change is increasing (becoming less negative).
Quick Reference Card
Screenshot and save this!
🔑 Must-Know Formulas
f(a + n·p) = f(a)
Period reduction — n any integer
f(x) = f(x − p)
Horizontal translation of itself
All solutions: x = a + n·p
If f(a) = k, infinitely many inputs give k
💡 Common Mistakes
❌ Confusing "positive" with "above midline"
Positive simply means the output is above the x-axis (y > 0). It has nothing to do with the midline.
❌ Forgetting negative n
n can be negative — add periods to reach positive range when input is negative.
❌ Concave up = increasing
Concave up means the rate of change is increasing — not the function itself!