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Topic 3.1 · Definition
What is a periodic relationship?
f(x + p) = f(x) for all x
What makes it 'periodic'?
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Topic 3.1 · Period
How do you find the period from a graph or table?
Look for when the pattern exactly repeats
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Topic 3.1 · Core Formula
If f has period p, how can you simplify f(x + n·p)?
f(x + n·p) = f(x)
n can be any integer — positive or negative
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Topic 3.1 · Midline
What is the midline of a periodic function, and how do you calculate it?
Midline = (max + min) / 2
It's a horizontal line halfway between max and min
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Topic 3.1 · Amplitude
What is the amplitude of a periodic function?
Amplitude = (max − min) / 2
Distance from midline to maximum
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Topic 3.1 · Positive vs Negative
When is a periodic function considered 'positive'?
Be careful — this is NOT about the midline!
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Topic 3.1 · Concavity
On a periodic graph, what does concave up vs concave down tell you about the rate of change?
Think about whether rate of change is increasing or decreasing
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Topic 3.1 · Worked Example
g has period 5. Known: g(4) = 7. Find g(14).
g(14) = g(4 + 2·5) = g(4)
How many full periods fit between 4 and 14?
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✓ Answer — Finding the Period
Find two identical points → measure horizontal distance
Look for peaks, troughs, or zero crossings. The horizontal distance between any two identical consecutive points = period p.
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✓ Answer — Periodic Relationship
f(x + p) = f(x) for all x
The function repeats every p units. p is the smallest positive value for which this holds — the least possible period.
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✓ Answer — Midline
y = (max + min) / 2
A horizontal line halfway between maximum and minimum output values. The function oscillates equally above and below it.
Example: max=7, min=3 → midline y=5
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✓ Answer — Core Formula
f(x + n·p) = f(x)
Adding or subtracting any whole number of periods leaves the output unchanged. n can be negative — strip periods to reach a known value.
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✓ Answer — Positive vs Negative
f(x) > 0 means output is above the x-axis
Positive = above y = 0 (the x-axis), NOT above the midline. This is a very common AP exam mistake — midline and x-axis are different!
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✓ Answer — Amplitude
(max − min) / 2
Distance from the midline to the maximum (or midline to minimum — same value).
Example: max=7, min=3 → amplitude = (7−3)/2 = 2
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✓ Answer — Worked Example g(14)
g(14) = g(4) = 7
14 = 4 + 2·5 (two full periods added). Adding periods doesn't change the output: g(4 + 10) = g(4) = 7
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✓ Answer — Concavity
Concave up → rate of change increasing
Concave up (curves like a bowl ∪): rate of change is increasing.
Concave down (curves like a hill ∩): rate of change is decreasing.