Sinusoidal Modeling

Pre-Calculus · Axiom Academy

Fit a sine or cosine curve to real periodic data by reading four numbers — midline, amplitude, period, and phase — straight off the highs and lows. 1. The Midline and the Amplitude Start with a city's daily temperature: a high of 85°F at 3 PM and a low of 65°F at 3 AM . Two numbers fall out immediately. The midline D is the average of the high and low — the level the curve oscillates around. The amplitude A is half the gap between them — how far it swings above and below that midline. Midline = average of max and min Amplitude = half the max-to-min swing The period is the time for one full cycle — measure it as the gap between two consecutive highs (or two consecutive lows). Temperature repeats once every 24 hours . The constant B is just how that period maps onto one full trip around the circle, so a longer period means a smaller B (a slower, more stretched-out wave). Peak-to-peak (or trough-to-trough) distance. Here that is 24 hours. A 24-hour cycle gives , a gently stretched wave. Halve the period and B doubles — the wave packs in twice as fast. The cycle length is 24 hours, so . This B is what stretches a bare or to span one day from peak to peak. A plain cosine peaks at t = 0 . But our data peaks at 3 PM , which is t = 15 hours after midnight. The phase shift C slides the whole curve sideways so its peak lands on the data's peak. Using cosine, we just set C to the time of the maximum: C = 15 .

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