AP Precalculus: Periodic Functions Cheat Sheet

The core ideas of AP Precalculus: Periodic Functions distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Amplitude

Half the vertical distance between max and min values.

Midline

Horizontal center line of a sinusoid, equal to the average of max and min.

Period

Horizontal length of one full cycle. For sin(Bx) or cos(Bx), period=2pi/|B|.

Phase Shift

Horizontal translation of the waveform caused by constants inside the input.

Inside vs Outside

Inside terms control horizontal behavior; outside terms control vertical behavior.

Key Terms at a Glance

Periodic Function:A function that repeats its values at regular intervals. f(x + T) = f(x) for all x, where T is the period.

Amplitude:The maximum displacement from the midline of a sinusoidal function. Amplitude = |A| in y = A sin(Bx) + D.

Midline:The horizontal line about which a sinusoidal function oscillates, located at the average of the maximum and minimum values.

Period:The smallest positive value T such that f(x + T) = f(x). For y = sin(Bx), the period is 2π/|B|.

Phase Shift:A horizontal translation of a periodic function. In y = sin(B(x - C)), the graph shifts C units to the right.

Frequency:The number of complete cycles per unit of the independent variable. Frequency = 1/period = |B|/(2π).

Sinusoidal Function:A function that can be written in the form y = A sin(B(x - C)) + D or y = A cos(B(x - C)) + D, producing smooth wave patterns.

Vertical Shift:A transformation that moves the entire graph up or down. In y = A sin(Bx) + D, the vertical shift is D units.

Angular Frequency:The coefficient B in y = sin(Bx), which controls how quickly the function completes its cycle. Related to period by B = 2π/T.

Reflection:A transformation that flips the graph across the midline (for negative amplitude) or across the y-axis (for negative B inside the function).

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