Reading a Wave's Equation Off Its Graph

A sine or cosine graph is built from the same unwrapped circle idea, but stretched and slid. The max and min tell you two things at once: how far the wave sits from the x-axis (midline d) and how far it swings above and below that line (amplitude a). The midline is the average of the max and min, d = (max+min)/2. The amplitude is half the total vertical distance, a = (max-min)/2. These two numbers alone fix the top-to-bottom shape of the wave.

The side-to-side shape comes from the period, the horizontal distance between two consecutive peaks (or two consecutive troughs). Since period = 2pi/b, you solve for b by b = 2pi/period. A shorter period means a larger b, meaning the wave repeats faster. Once you know where a peak actually sits, you can line up sine or cosine's own peak with it to find the phase shift c, since sine naturally peaks a quarter-period after it crosses the midline going up, while cosine peaks right at x = 0 before any shift.

This topic is the reverse of graphing from an equation: here you start from the picture and recover a, b, c, d. It sits right before combining these features with the unit circle's exact values, and before switching between sine and cosine descriptions of the same wave, which is really just a horizontal shift in disguise.