Amplitude, Period, Phase Shift and Midline (F 705)

Every wave the exam draws is a stretched, squeezed, slid, and raised version of the basic sine or cosine curve. The coefficient a out front is the amplitude: it stretches the wave's height, so the curve runs from k - a up to k + a. The number b inside controls how fast the wave repeats. Since one full trip around the unit circle is 2pi radians, the input bx completes that trip when x = 2pi/b, so the period is 2pi/b. A bigger b squeezes more cycles into the same stretch of x-axis — that is why y = sin(3x) repeats every 2pi/3, not every 6pi. The 3 speeds up the angle, it does not slow the wave down.

The constant k shifts everything up or down and becomes the midline, the horizontal line the wave oscillates around. The trickiest piece is phase shift, because it only shows up correctly once b is factored out front of the parentheses. Written as sin(bx - c), the shift is hidden; you have to rewrite it as sin(b(x - c/b)) to see that the graph has moved c/b units, not c units.

These four numbers turn a bare equation into a full picture: start at the midline, bounce up to the max by a, trace one period, and land back down. This same reading skill carries forward into matching equations to graphs and into the law of sines and cosines questions, where recognizing a wave's shape quickly saves real time.