Maths: Analysis and Approaches · Geometry and trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The graphs of sin x and cos x are waves that repeat every 2π and stay between −1 and 1. The graph of tan x repeats every π, has no maximum or minimum, and has vertical asymptotes where cos x = 0.
The amplitude is the distance from the middle line of the wave to a peak. The period is the length of one complete cycle.
In f(x) = a sin(b(x + c)) + d each letter has one job: a sets the amplitude, b sets the period, c slides the graph sideways and d moves the middle line up or down. The same letters describe tides, big wheels and temperatures, where d is the average value and a is the greatest change from it.
Students give the period of y = sin 2x as 4π. A larger b squeezes the wave, so divide: the period is 2π ÷ 2 = π.
Answer: Period π/2; minimum value 3, first reached for positive x at x = π/2.
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