Exploring y = a sin(bx + c)

by Thuy Nguyen

In the equation y = a sin(bx + c), we call |a| the amplitude of the graph, c the phase angle, 2p/|b| the period or wavelength, and –c/b to be the displacement of the sine curve. We obtain the displacement formula by solving for x in the equation bx + c = 0, so the displacement is basically the quantity that the curve is moved in a horizontal direction from it’s normal position. If the displacement is negative, the curve will be shifted to the left, and to right otherwise. The period is the distance it takes for the sine curve to start repeating again.

We’ll first look at the graph of y = sinx:

We see here that the domain of this function is the set of all real numbers, and the range is the set [-1,1]. We’ll now examine how different values of a, the amplitude, can affect the above function. Let’s look at the graphs for the functions

y₁ = 3 sinx

y₂ = ½ sinx and

y₃ = - 2 sinx, in blue, red, green, respectively.

We note that for the function y₁ = 3 sinx, the range is now [-3,3]. For the functions y₂ and y₃, the ranges are now [- ½, ½ ] and [-2, 2], respectively. In fact, the range for y = a sin(bx + c) is just [-|a|, |a|].

Now we’ll examine how different values of b can affect the original sine graph. Since the period of the sine function is 2/b and the displacement is –c/b, we can already see that values of b will greatly affect these two things. But for now, we’re taking c to be zero, so we’ll not worry about displacement just yet. So now let us look at the graphs for the functions

y₁ = sin(2x)

y₂ = sin( ½ x) and

y₃ = sin(-3x), in blue, red, green, respectively.

The periods for y₁, y₂, and y₃ are 4 and 2/3 respectively. We then see that the greater the value of b, the lesser the frequency.

Let’s us now examine the graph of y = sin(4x + 2). We see that

a = 1, b = 4, c = 2

So the displacement is – 2/4 = - ½ and the period is 2/4 = /2.

First look at the graph of y = sin(4x):

Since the displacement is – ½, we can correctly assume that if we shift the above graph to the left by a distance of ½, we will get the graph y = sin(4x + 2). (in red)

Now let’s see what happens when we change the phase angle to – 6:

So the above graph is the graph of the function y = sin(4x – 6). As the displacement is 3/2, we see that the graph has been shifted to the right by a distance of 3/2 when compared to the function y = sin(4x). What values of c would align the two functions y = sin4x and y = sin(4x + c) so that they have the same graph? Well, we want values of c such that –c/4 = n/4. Therefore, we need c = - np.

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