Matching a Trigonometric Graph to Its Equation

A sine or cosine graph is just a repeating wave, and every feature of that wave corresponds to a number in y = A sin(Bx) + k or y = A cos(Bx) + k. The midline is the horizontal line exactly between the peaks and troughs — that height is k. The amplitude A is the distance from the midline up to a peak, or equivalently half the distance from peak to trough. Neither of these needs any formula sheet; they're just measurements you read straight off the graph.

The period is trickier because it hides inside B. The period is the x-distance between two consecutive peaks (or two consecutive troughs), and it equals 2pi/B. So once you measure the period off the graph, you solve for B by dividing 2pi by that distance. Whether you use sine or cosine as your base function depends on where the wave sits at x = 0: start at the midline heading up, use sine; start at a peak, use cosine. This is exactly the same ratio logic as the unit circle's x and y coordinates, just stretched into a continuous picture as the point spins around.

This skill sits right after you've learned what sine and cosine look like as graphs, and right before transformations get stacked — shifts, reflections, and period changes all at once. A flipped wave, one that starts at a trough instead of a peak, is the same base function with a negative sign in front of A. Matching equations to graphs is really the reverse of graphing: instead of plotting from an equation, you're reading the equation's numbers back out of a picture.