Carisa Lindsay

Assignment 1

Exploration of
Functions

This is an
exploration of the graph as
*a*
increases:

It would seem that *a* = 1 is the simplest
version of this graph because it is free of any transformations.

*a *= 0

As *a* increases, the
ÒloopÓ in quadrants I and IV seems to narrow though it maintains a ÒlengthÓ of
one unit. Furthermore, as *a* increases,

*a* = 3

*a* = 5

*a* = 10

As *a* increases, the
ÒloopsÓ in quadrant II and III extend and become narrower. Furthermore, the ÒloopÓ in quadrants I
and IV also narrows but still maintains a length of unit one.

However, if we
decrease the value of *a*, to a value less than zero, the graph reverses
quadrants. The shape is similar
except that the ÒloopsÓ are in different quadrants. The smaller (i.e., more negative) *a* becomes, the larger
the ÒloopsÓ in quadrants I and IV become.
The significant difference with *a* being negative appears to be the
absence of the additional loop- it would seem this missing loop should be in
quadrants II and III now.

*a *=-10

*a *=-5

*a*=-3

Now we can take a
closer look at what happens to the graph for values of *a* between zero and
one. As *a* approaches 1, the
ÒloopsÓ in quadrants II and III become less significant and eventually
disappear at *a*
= 1.

* *

*a *=.1

*a* = .3

*a* =.5

* *

However, for values
of a between 0 and -1, there are not many noticeable differences comparing
these graphs. The most obvious
difference between these values and the positive *a* values is the
absence of the ÒloopsÓ in quadrants II and III, much like other negative values
of *a*.

*a* =-.5

*a* =-.1

There are, however,
only marginal differences between the values of *a* = 0 and *a* = -1. As *a* becomes more
negative, the ÒloopsÓ become ÒwiderÓ but all maintain unit length. In the following graph, the red is *a* = -0.3, pink is *a* = -0.5, blue is *a* = -0.7, and green is
*a*
= -0.9.

As you can see in the
following animation, as *a* increases, the ÒloopsÓ become more
pronounced and change quadrants.
All share the unit length of one in Quadrants I and IV.