Graphing Trig Functions — Amplitude and Period

ACT Math · Axiom Academy

In , one number sets how tall the wave is and another sets how wide one cycle is — watch each control its own dimension. 1. Amplitude — the coefficient a stretches the height Start with the plain wave , which rises to 1 and falls to -1 . Multiply by a and every height scales by that factor: the peak jumps to |a| and the trough to -|a| . The amplitude is the distance from the midline up to a peak — it is exactly |a| . Amplitude = distance from midline to peak The wave oscillates between d - |a| and d + |a| : amplitude = |a| = |3| = 3 , so the wave reaches a max of 3 and a min of -3 . : amplitude = |a| = |-5| = 5 . The negative sign flips the wave upside down, but the amplitude is still 5 — always positive. 2. Period — the coefficient b sets the width The number b inside the function speeds the wave up. One full cycle takes units: a larger |b| compresses the wave into a shorter period, a smaller |b| stretches it out. Below, fits two complete cycles into the same width that needs for one. The wave is compressed; more cycles fit in the same width. has period . The wave is stretched out. has period . Read off the coefficient of x inside the function before you divide. Period . The height (amplitude) is untouched by b . : period — a full cycle every units, i.e. 4 cycles in . : here , so period — a very stretched-out wave. 3. Putting it together — read the equation off a graph

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