EMAT 6680 Assignment 1: Activity 5
Examine the graphs of y= a sin(bx+c) for different values of a, b, and c.
Each parameter of a function, such as y= a sin(bx+c), individually affects the graphical components. By looking at the three different animations we can see how the three included parameters, a. b, and c affect the sinusoidal graph. As most often presented to students, a=1, b=1, and c=0 we have the standard y= sin(x) graph. This graph is illustrated below.
For this basic graph our y-intercept is 0, our period is 2pi, and our amplitude is 1. The y-intercept is defined as the value where the graph crosses the y-axis. The period is defined as the horizontal distance it takes for the graph to complete one cycle. The amplitude is defined as the vertical distance between the minimal point and maximum point of the graph, then dividing that value by two. Once we begin to adjust the variable values our original sine function graph begins to be changed as well.
First, let's look at what happens to our graph as we vary the value of a. The coefficient to our function directly affects the amplitude of the graph. Amplitude, as previously mentioned, is half of the vertical distance between the minimal and maximum peaks of the graph. If our variable a = 5, then the amplitude is also 5. The most interesting part of varying the parameter a is how the graph changes as it becomes negative. The amplitude value corresponds to the absolute value of a. Thus, the negative portion of the coefficient does not affect the amplitude value, but does still affect the graph. When this coefficient is negative, the graph is altered by reflecting across the x-axis. Any y-value that was positive is now negative, and any y-value that was negative is now positive. Below there is a representation of the graph y = a sin(bx+c) as the value of the coefficient a varies between -10 and 10. (The value of b remains 1 and the value of c remains 0)
Varying parameter a in graph: y= a (sin (bx+c)):
Amplitude of Sine Graph
Adjusting the value of parameter b changes the period value of the graph. The period of a trigonometric function is again defined as the horizontal distance required for a curve to cycle through one time. To determine the period value, you must divide 2pi (the original period value of a curve) by the value of the parameter b. Therefore, as the value of b increases the period value decreases. We can also look at this as a horizontal compression or stretch since the period is a change in the horizontal value for one cycle. This accordion-like change in the graph can be seen as we vary the value of parameter b. In addition, by taking the absolute value of b, we eliminate any affects of a a negative value. The video below illustrates the sine function graph as b varies between the values of -10 and 10. (The value of a remains 1 and the value of c remains 0)
Varying parameter b in graph: y= a (sin (bx+c)):
Period of Sine Graph
Lastly, we can look at the sine curve and examine the changes as the value of the variable c varies. Since the parameter is included inside the parenthesis, it is obvious that as c is adjusted the sine curve is adjusted horizontally from its original position. This is referred to as a graph's phase shift. To determine a sine functions phase shift we must divide the values -c/b. This allows us to see that the parameters b and c together affect this attribute of the graph. When -c/b produces a positive value, the phase shift is positive and moves the curve to the right. If -c/b produces a negative value, then the phase shift is negative and moves the curve to the left. The illustration below shows the sine function as the variable c varies between the values -10 and 10. (The value of a remains 1 and the value of b remains 1)
Varying parameter c in graph: y= a (sin (bx+c)):
Phase Shift of Sine Graph
From the included video illustrations, it becomes easy to see that each parameter included in the sine function greatly affects the sine graph. Watching the parameter values vary gives us a better opportunity to visually recognize these graphical changes and become more aware of the sine graph as a whole.