Sine wave explorer
Move one slider at a time and watch the blue wave y = a sin(b(x − c)) + d. The grey wave is plain y = sin x, for comparison. Each letter does one job: a sets the height, b the speed, c the slide, d the lift. When you know the four jobs, press New wave to match and copy the orange wave.
What's going on
Amplitude is |a|: how far the wave rises above its middle. A negative a turns the wave upside down but the amplitude is still positive. Midline is y = d: the horizontal line the wave is centred on. So the highest value is d + |a| and the lowest is d − |a|.
Period is 2π/b: the length of one full wave. Plain sin x takes 2π to repeat. A bigger b runs through the cycle faster, so the wave is squashed and the period is shorter. Phase shift is c: the wave slides right by c (left if c is negative). The orange dot marks where a cycle starts: on the midline at x = c.
To read an equation off a graph, work in this order. Midline: halfway between the top and the bottom. Amplitude: from the midline to the top. Period: from one peak to the next, then b = 2π ÷ period. Shift: find a place where the wave crosses the midline going up, and that x is c.
One wave has many equations. Sliding by a whole period changes nothing, and flipping (a → −a) is the same as sliding half a period. The match challenge accepts any of them.
Sheets: Reading a sine wave and Predicting the tide.