# 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).