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# 1. Amplitude, Phase Shift, and Period

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: