This picture shows a graph of \(\sin(x)\).
The amplitude of a sine wave is determined by the coefficient in the front. An amplitude greater than \(1\) will make the waves taller while an amplitude between \(0\) and \(1\) will make the waves flatter. An amplitued of \(0\) is a flat line, and negative amplitudes flip the waves around. Here is a picture of \(2\sin(x)\), which has an amplitude of \(2.\)
The phase shift of a sine wave is determined by the number subtracted from \(x\) within the sine function. The phase shift will move the wave left or right. A phase shift of \(2\pi\) is the same as a phase shift of \(0\). Here is a picture of \(\sin(x-1)\) which has a phase shift of \(1.\)
The period of a wave is determined by the coefficient of \(x\) within the sine function. A higher period means more peaks within an interval. Here is a picture of \(\sin(2x)\) which has period \(2.\)
The vertical shift of a wave is determined by the number added to it. Adding positive numbers move the wave up and adding negative numbers moves the wave down. Here is a graph of \(\sin(x)-1\) which has a vertical shift of \(-1.\)
Different waves can be created by changing the amplitude, phase shift, period and vertical shift. This is the graph of \(2\sin(2(x-1))-1.\) It has amplitude \(2,\) phase shift \(1,\) period \(2,\) and vertical shift \(-1.\)
You can change the graph of the cosine function here: