by : Tim Lehman

Question #5: Examine the graphs of y=a sin(bx + c), for different values of a, b, and c.

One of the more basic graphs of this form is y=sin(x). To see this graph, click here. First, we will examine a, b, and c each separately. When we fix each of the other two, it will allow us to get a better idea of each variable.

Examination of variable a: Variable a affects the magnitude of the graph. This makes sense because a simply raises or lowers y directly because it is multiplied by sin(bx + c). Further, we can see that, when a is not equal to zero, our original equation leads to y/a = sin(bx+c). If sin(bx+c) is held constant, y must increase as a increases. We can see this in the graph of y=a sin(x), when a varies from -10 to 10. To see this graph, click here.

Examination of variable b: Variable b controls the amplitude of the graph. It expands the graph multiplicitively. The variable b affects the rate of change among x-values. This is evident in the graph of y=sin(bx), where b varies from -10 to 10. To see this graph, click here.

Examination of variable c: Variable c shifts the graph to the right or left. It leaves the amplitude and magnitude of the graph unchanged. The c-value affects the y-intercept of the graph. There exists some c such that y=sin(x+c) is the same as y=cos(x). However, the control of variable c is best seen in the graph of y=sin(x+c), where variable c varies from -10 to 10. To see this graph, click here.

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