y = asin(bx + c)

By Karl Mealor


We will examine the function y = asin(bx + c) for different values of a, b and c. We will begin by graphing y = sinx (a, b, c = 1). The remainder of the graphs will be compared to this form of the function:

y = sinx



The next few graphs demonstrate what happens when a is varied. Notice that the amplitude of the curve (distance of the maximum and minimum points from the x-axis) varies according to the change in this coefficient.

y = sin x

y = 3sinx

y = 0.5 sinx


Notice that when a = 3 (graphed in red), the amplitude of the curve is 3. Likewise, when a = 0.5 (graphed in blue), the amplitude is 0.5.


It is worth noting that if a is negative, the graph is reflected across the x-axis. This is demonstrated by the following graphs of y = 2sinx and y = -2sinx.

y = sinx

y = 2sinx

y = -2sinx

Notice that the graph of y = -2sinx (graphed in blue) has an amplitude of 2 and is the reflection of y = 2sinx (graphed in red).


In conclusion, the amplitude of the sine function in the form

y = asin (bx + c)

is determined by the absolute value of a .




We will now demonstrate the effect of varying b while a and c remain constant. The following graphs indicate the result of letting b = 1, 2 and 0.5 while a = 1 and c = 0:

y = sinx

y = sin2x

y = sin0.5x

It is evident that changing b affects the period of the sine function. Specifically, the period of each function is equal to the absolute value of 2p/b. Notice that when b = 1 (graphed in pink), the period is equal to 2p/1 = 2p, or about 6.3. When b = 2 (graphed in red), the period equals 2p/2 = p which is about 3.1. When b = 0.5 (graphed in blue), the period equals 2p/0.5 = 4p or about 12.6.


To summarize, the period of the sine function in the form

y = asin (bx + c)

is determined by the absolute value of 2p/b .




Phase Shift

We will conclude by investigating the effect of changing c while holding a and b constant. In the following graphs, a and b will remain equal to 1 while c = 0, 1 and -2.

y = sinx

y = sin(x + 1)

y = sin(x - 2)

Changing c effectively shifts the graph to the left or to the right. This phase shift is determined by -c/b. For example, when c = 1 (graphed in red), the graph has the same period and amplitude as when c = 0 (graphed in pink), but has been shifted -(1/1) = -1, or 1 unit to the left. When c = -2 (graphed in blue), the graph also has the same amplitude and period as the graph of y = sinx, but it has been shifted -(-2/1) = 2 or 2 units to the right.


To summarize, the phase shift of the sine function in the form

y = asin (bx + c)

is determined by -c/b .