Exploring graphs of y = a sin(bx + c)

Consider what hopefully is the familiar graph of the equation y = sinx below.

We see that this is a periodic function with period 2 pi, amplitude of 1, and passes through the origin.

We note that this equation is a special case of the general form: y = a sin(bx+c). In the graph above a = 1, b = 1, and c = 0. A natural question arises. How do different values for a, b, and c affect the graph?

First, we'll look at varying a by fixing b and c at 1 and 0 respectively.

So we see that varying a changes the amplitude of the sine curve but does not change the period or position of the curve. In fact, we see that the value of a is the amplitude of the curve. So what about negative values of a? A negative value for a produces a graph that is a reflection of the graph for the corresponding positive value for a over the x-axis. This is illustrated below.

Next we will examine different values for b by fixing a and c at 1 and 0 respectively.

So now we see that changing b changes the period of the sine curve alone. In fact, we see that the period is inversely related to the value of b. When b = 1/2, the period of the curve is twice the period when b = 1. Further, when b = 3, the period is 1/3 the period when b = 1.

We again find that a negative value for b simply produces a graph that is the mirror image across the x-axis from the corresponding positive value for b. We illustrate this below.

Finally, we will look at varying the value for c and see what effect it has on our graphs. We will again fix a and b at 1 and 1 respectively.

Now we see that changing c affects the horizontal position of the sine curve. We can see that positive values of c move the sine curve c units to the left while negative values of c move the curve c units to the right.

So lets put what we have learned to the test. Given the following equation,

y = 3/2(sin(2x-2))

what can we say about the amplitude, the period, and the position of the graph relative to the graph of the equation y = sinx?