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.
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 = .3
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.
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.