## Families of Trig Functions

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

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

A,B,C,Dare theparametersof the family. To keep things simple for now, we will set the parameterCequal 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 sinceA,BandDare all initially set to 0. Start by using the sliders to makeAequal to 1 andBequal to 1. You will then see the graph ofy = sin(x). Then answer the following questions.1. Starting with

A=1increaseAto 5 by using the slider and describe what happens to the graph. Then decreaseAto 0 and then to -5. Describe what happens to the graph.2. Restore the value of

Ato 2 and varyB. Describe what you see. In particular, describe what happens whenBchanges from positive to negative.3. Remember (or look up!) what is meant by the

periodof the functionf(x). Use the graph to estimate the period whenB =3and whenB=0.5. Then try some other values. Roughly speaking, how doesBaffect the period? (You will learn from your text that there is very precise relationship between the period and the value ofB. )4. This one's easy. You should be able to guess the effect of changing

D. Confirm your guess by varying the value ofDusing its slider.We now want to study the remaining parameter,

C. We will dispense withDand change the applet so that it can control the parameterC. 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 thatA = 1andB = 1. Initially,Cis set to 0. Increase it's value to 1, then 2, then 3 and describe what happens to the graph. Then setCback to 0 anddecreaseit 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 inC?2. Reset

Cto 0 and changeBto 2. Repeat the experiments in the preceding problem. By how many units does the graph appear to move with each one unit change inCwhenBequals 2?3. Reset

Cto 0 again and changeBto 0.5. Repeat the experiments in the preceding problem. By how many units does the graph appear to move with each one unit change inCwhenBequals 0.5?4. Summarize your observations from the preceding three problems, to describe roughly how changing

Caffects the graph, and how that change depends onB. (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.