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.