Recipe

Graph a sinusoidal function from its equation

Use when: you're given y=A\sin(B(x-C))+D or y=A\cos(B(x-C))+D (or an equation that can be rewritten into that shape) and asked to sketch or describe its graph

Goal: the midline, amplitude, period, and phase shift, plus enough plotted points to sketch one full cycle

Taught in Amplitude, period, and phase shift

1. 1

   Read A and D straight off the equation. Amplitude is |A|; the midline is the horizontal line y=D. The max is D+|A| and the min is D-|A|.

   Example

   For y=3\sin\left(2\left(x-\frac{\pi}{4}\right)\right)-1: A=3, D=-1, so amplitude 3, midline y=-1, max 2, min -4.

   Why: A stretches the wave vertically and D slides the whole wave up or down; neither one touches timing at all, so they're safe to read off before anything else.

2. 2

   If the inside expression isn't already in B(x-C) form, factor B out of it first. If B itself is negative, rewrite using \sin(-\theta)=-\sin\theta or \cos(-\theta)=\cos\theta before continuing. Once B is a clean positive number, the period is T=\dfrac{2\pi}{|B|}.

   Example

   For y=\sin(2x-\pi): factor first, 2x-\pi=2\left(x-\frac{\pi}{2}\right), so B=2 and T=\dfrac{2\pi}{2}=\pi.

   • the parent is \sin and B is negative — flip the sign of A instead, and use |B| from here on
   • the parent is \cos and B is negative — the negative sign simply vanishes; use |B| from here on

   Why: T depends only on B, but reading B straight off an unfactored or negative-signed equation also corrupts the phase shift in the next step, so cleaning it up now protects both.

3. 3

   Once the inside is in B(x-C) form, C is the phase shift: read it directly. A minus sign inside the parentheses shifts the graph right by C; a plus sign shifts it left.

   Example

   Continuing y=\sin\left(2\left(x-\frac{\pi}{2}\right)\right): C=\frac{\pi}{2}, so the graph shifts right by \frac{\pi}{2}.

   Why: this is Chapter 4's inside-lies rule applied to a wave: the sign sitting inside the parentheses describes the opposite direction from what it looks like.

4. 4

   Mark the midline y=D. Starting at x=C, plot five landmark points spaced \dfrac{T}{4} apart, not T apart, using the parent function's own starting shape to decide what each point is.

   Example

   For y=3\sin\left(2\left(x-\frac{\pi}{4}\right)\right)-1: T=\pi, so the five points sit \frac{\pi}{4} apart starting at x=\frac{\pi}{4}: midline \left(\frac{\pi}{4},-1\right), max \left(\frac{\pi}{2},2\right), midline \left(\frac{3\pi}{4},-1\right), min (\pi,-4), midline \left(\frac{5\pi}{4},-1\right).

   • parent is \sin — midline (climbing), max, midline (falling), min, midline (climbing)
   • parent is \cos — max, midline (falling), min, midline (climbing), max

   Why: the five landmarks sit at quarter-period spacing, not full-period spacing — the exact slip this section's own common-mistake box warns about.

5. 5

   Repeat the same five-point cycle leftward and rightward from the plotted period, one full period T at a time, for as far as the sketch needs to go.

   Why: sine and cosine, and every transformation of them, repeat for every real input, not just the one window plotted.