Assignment 1 :: Graphs

Exploring the function of y = a sin bx + c

By Jamie K. York

The sine wave is a function that can be seen in a variety of areas, including mathematics, music, physics, and engineering. It's basic form is as follows:

 y = a sin (bx + c)   Each of the given variables, as they are modified, have a specific impact to the resulting sine wave, or graph. These variables are explored in further detail below. In this exploration, the graph of y = sin x will be the starting point, from which various transformations can be discovered. Note that this graph is represented by a = 1, b = 1, and c = 0.

Amplitude (a)

The amplitude of the graph, often denoted as A, represents the maximum distance from the center of the sine wave. In exploring this variable, 0 may be used for A. This simplifies the function, eliminating the sine function and resulting in a function of y = 0.

The base function, y = sin x, has an altitude of 1. Exploring the definition and graph above, it shows that the maximum distance from the center of the wave is 1. This can be seen both above and below the x-axis. Taking this idea and expanding upon it, additional amplitude values can be explored as displayed in the video below.

To take a more focused look at the amplitude, we can explore some specific values. Below are the graphs representing the standard form with 5 different values: -2, -1, 0, 1, 2.

Angular Frequency (b)

The angular frequency of the graph, often denoted as ω, represents the number of periods within an interval of length 2¹. Notice that as b increases from 0, it appears to shrink wave without changing the height. On the other hand, as b approaches 0, the wave appears to stretch.

To take a more focused look at the angular frequency, we can explore some specific values. Below are the graphs representing the standard form with 5 different values: -2, -1, 0, 1, 2. You can again see the conclusions drawn earlier about the shrinking and stretching of the wave, however, we can also see that +b and -b are reflective over the x-axis.

Phase (c)

The phase of the graph, often denoted as θ, represents the positioning of the wave. As c increases, the wave transitions down the positive end of the x-axis. As c decreases, the wave does the opposite and transitions to the negative end of the x-axis.

To take a more focused look at the phase, we can explore some specific values. Below are the graphs representing the standard form with 5 different values: -2, -1, 0, 1, 2. The exact value of c represents where our original function of y = sin x is transitioned to on the x-axis. For example, a value of b = 2 moves the original x-axis intersect at 0 to 2. Likewise, for b = -2, it is transitioned to -2 on the x-axis.

Further exploration

We can revisit the original graph to explore the radian values that are relevant to it. Focusing on one period of the wave, or a length of 2¹. This length contains one sine wave, which is then repeated indefinitely. The wave intersects the x-axis at the values of -¹, 0, and ¹. The wave continues to intersect the x-axis at every interval of ¹ (2¹, 3¹, 4¹, etc.).

Consider an additional variable that we will represent as d, resulting in the following equation:

y = a sin (bx + c) + d

*  What effect does the new variable d have on the graph?

*  What is the difference between a negative and positive value of d?

*  Does this variable modify the altitude, angular frequency, or phase of the graph?