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 “loop” in quadrants I and IV seems to narrow though it maintains a “length” of one unit.  Furthermore, as a increases,

a = 3

a = 5

a = 10

As a increases, the “loops” in quadrant II and III extend and become narrower.  Furthermore, the “loop” 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 “loops” are in different quadrants.  The smaller (i.e., more negative) a becomes, the larger the “loops” 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


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 “loops” 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 “loops” 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 “loops” become “wider” 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 “loops” become more pronounced and change quadrants.  All share the unit length of one in Quadrants I and IV.



Click here to see animation




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