Families of Trig Functions

The basic sine function f(x) = sin(x) belongs to a family of functions

f(x) = A sin(Bx-C) + D

where A,B,C,D are the parameters of the family. To keep things simple for now, we will set the parameter C equal to zero and consider only functions of the form

f(x) = A sin(Bx) + D.

Use the sliders on the applet to the left to understand how the parameters affect the shape of the graph of the function f(x). At first you will see only a horizontal line since A,B and D are all initially set to 0. Start by using the sliders to make A equal to 1 and B equal to 1. You will then see the graph of y = sin(x). Then answer the following questions.

1. Starting with A=1 increase A to 5 by using the slider and describe what happens to the graph. Then decrease A to 0 and then to -5. Describe what happens to the graph.

2. Restore the value of A to 2 and vary B. Describe what you see. In particular, describe what happens when B changes from positive to negative.

3. Remember (or look up!) what is meant by the period of the function f(x). Use the graph to estimate the period when B =3 and when B=0.5. Then try some other values. Roughly speaking, how does B affect the period? (You will learn from your text that there is very precise relationship between the period and the value of B. )

4. This one's easy. You should be able to guess the effect of changing D. Confirm your guess by varying the value of D using its slider.

We now want to study the remaining parameter, C. We will dispense with D and change the applet so that it can control the parameter C. To do this, click here: A sin(Bx - C).

(If you should want to return to the original, click here: A sin(Bx)+D)

Now let's concentrate on C. 1. Adjust the sliders so that A = 1 and B = 1. Initially, C is set to 0. Increase it's value to 1, then 2, then 3 and describe what happens to the graph. Then set C back to 0 and decrease it value to -1, then -2, then -3. Describe what happens then. By how many units does the graph appear to move with each of the one unit changes in C?

2. Reset C to 0 and change B to 2. Repeat the experiments in the preceding problem. By how many units does the graph appear to move with each one unit change in C when B equals 2?

3. Reset C to 0 again and change B to 0.5. Repeat the experiments in the preceding problem. By how many units does the graph appear to move with each one unit change in C when B equals 0.5?

4. Summarize your observations from the preceding three problems, to describe roughly how changing C affects the graph, and how that change depends on B. (You will learn from your text that there is a very precise way to express all this.)

Precalculus Materials by B. Kaskosz and L. Pakula, University of Rhode Island, Copyright 2002.