As b changes...

Recall that in the original sine curve y=sin x, a=1, b=1, and c=0. Let's explore what happens as the value of b changes. Consider the following examples.


Focus on comparing the purple and the red curves. The purple curve is simply y=sin x. The red graph, y=sin(2x) demonstrates what happens when b=2.

Define a period as the length it takes the curve to repeat the "wave cycle" of one crest and one trough. In y=sin x, one cycle starts at the origin. Continuing to the right along the x-axis, the cycle starts over again at the point x = . Thus, the period of y=sin x is .

In the graph of y=sin(2x), one cycle starts at the origin, but it ends at the point x=. Thus, the period for y=sin(2x) is .

By examining the remaining examples we can determine that the period for a sine curve will be .

Thus, for the graph of y=sin(0.5x), the period will be because b=1/2.

What happens when b is negative? Take a look at the graph of y=sin(-x). The curve is flipped or reflected over the x-axis. So, in general, the graph of a sine curve with a negative value of b is the graph of:reflected over the x-axis.

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