Advanced Algebra
Trigonometric Functions
See how circular motion makes periodic functions: read sine, cosine and tangent from the unit circle and from right triangles, measure angles in degrees and radians, graph and transform the waves, and solve trig equations with their inverses.
THE SYLLABUS
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Periodic Phenomena
Many things in the world repeat in a predictable pattern. A periodic function repeats its outputs, so one cycle tells you what comes next.
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The Period of a Function
The period P of a periodic function is the shortest distance after which its outputs repeat: f(x + P) = f(x) for every x.
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Horizontal Shifts of Periodic Functions
Shifting the graph of a periodic function left or right by its period P, or by any integer multiple of P, leaves the graph unchanged, so one period is enough to build the whole graph.
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Midline of a Periodic Function
The midline of a periodic function is the horizontal line halfway between its greatest and least outputs: M = (max + min) / 2. Shifting the graph up or down moves the midline by the same amount.
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Amplitude of a Periodic Function
The amplitude A of a periodic function is how far it swings up and down from its midline M: A = (max - min) / 2, so its outputs run from M - A up to M + A.
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Predator and Prey
Predator and prey populations rise and fall with the same period: while the prey is higher than its midline the predators increase, and while the predators are high the prey decreases, so the predator's peaks come a quarter period after the prey's.
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Level Check
Fifteen questions from Level 1: repeats, periods, far values, midlines and amplitudes, shifted and stretched graphs, and five graphs read for where they rise, fall, dip below zero, bottom out and reach a value.
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