Assignment 1 Sandy Cederbaum

**Beginning
with the values a=1, b=1, and c=1, we graph the
basic function y=sinx. This function has an Amplitude of 1 unit,Period a of ****, and a Phase Shift of 0 units. We will become more
familiar with these terms in this exploration so hang tight.**

**Let
us begin by exploring the effects of using different values of
a in the function
y=asinx. Click here in order to see
what happens to the graph of y=asinx as the value of a changes over the set of real
numbers ranging from -8 to 8. As we can see, the amplitude of the graph
changes accordingly.**

**What
appears to have happened to the graph of y= asinx as compared to y=-asinx? Look at
the following example where we compare the graph of y=3.2sinx
and y=-3.2sinx on the same set of axes. Conclusion.**
**Both of the graphs below
are said to have an amplitude of 3.2 (the absolute value of a).**

**What happens to the
graph when a=0?**

**Sketch the graphs:
y=0.5sinx , y=4.2sinx , and y=-7sinx.**

**Let
us next explore what happens to the function y=sin(bx) when we allow
b to vary over
the set of real numbers from -8 to 8. Click here in order to see
how the graph is effected. Write a few sentences to describe the
effect that the changing b values have on the graph of y=sin(bx) for 0<b<1 and for
b>1 if we define
the period of the sine function
to be the number of units (degrees or radians) in one complete
cycle of the graph.**

**Did
you conclude that the period of the graph is effected by the changes
in the value of b?**

**How
can we find the period of the graph of y=sin(bx)?**

**Sketch the graphs of
**,

**What appears to have
happened to the graph of y=sin(bx) as compared to y=sin(-bx)? Look at the following example
where we have graphed y=sin(-1x)**
**and y=sin(1x) on the same set of axes.**

**Here the sine function
is deceiving because we could interpret our results as a vertical
flip (a flip over the x-axis). However, if we do a little analysis
and/or plug in some reasonable values for x in both equations
(try x=, ,
and ), we see that opposite x values
return the same y values. Thus we actually have a "horizontal
flip". We can think of it as flipping the graph over the
y-axis.**

**Let
us next explore the graph of y=sin(x+c) as c varies. Click here to see an animation
of the graph as the value of c varies from -5 to 5. Write a few sentences
to describe the effect that the changing c values have on the graph of y=sin(x+c). This horizontal
shifting of the graph is called a phase shift. Can you find the distance and
direction of the phase shift for the graphs of and
y=sin(x-7)? Here are the answers.**

**Now
let us see what happens when we put all of the pieces together
and vary the values of a, b, and c. Graph the following:**

**Congratulations.
You have just completed a study of transformations of the graph
of y=asin(bx+c). You should feel fairly comfortable sketching
a fairly accurate graph of any awkward sine function using what
you now know about the Amplitude, Period and Phase shift of y=asin(bx+c).
As an extension of this exploration, you may want to consider
what happens to the graph when we add a constant (y=d+asin(bx+c)).**