Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.
Write a sinusoidal model from maximum, minimum, period, and starting-position clues
Problem
A sinusoidal quantity has maximum \(9\) at \(t~=~2\) and minimum \(1\) at \(t~=~8\), then repeats. Use the diagram to derive its amplitude, midline, period, and phase, and write a cosine model.
Build the model from vertical and horizontal evidence separately. The extreme values determine amplitude and midline, while the time from a maximum to the next minimum is only half a cycle and must be doubled for the period. Anchor a cosine phase at the known extreme, convert period to angular coefficient, and verify both landmarks in the finished model.
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