Introduction

In this write-up, I examine the graphs of the graphs of  using the different values of a. The focus is how the graph of  appears as a changes.

 The graph of  is a parabola with the vertex at the origin(0 , 0).

1) Let's notice the y values .
All the y-values is positive. The squaring operation makes the result. Also, the parabola never touches the x-axis except at the origin(0, 0).
 

2) Now, let's look at the shape of the graphs.
 The parabola open up and has a minimum, when "a" is positive.
 
 

3) Finally, notice that the

The graph of     is wider than the graph . Obviously,  is wider than  .
Therefore. we can see that the graph becomes more narow as the value of a increases, when a>0. Similary, the graph  becomes wider as the value of a decreases, when a>0.
 
 

Again, the graph of , a<0  is a parabola with the vertex at the origin(0 , 0).

1) Let's notice the y values .
All the y-values is negaitive. Also, the parabola never touches the x-axis except at the origin(0, 0).
 

2) Now, let's look at the shape of the graphs.
The parabola opens down and has a maximum, when a is negative.
 

3) Finally, notice that the

The graph of  is wider than the graph   . Obviously,  is wider than  .
Therefore. we can see that the graph becomes more narow as the value of a increases, when a<0. Similary, the graph  becomes wider as the value of a decreases, when a<0.

Let's compare of the following graphs.
 
 

For the same x-value, y-absolute values of both of graphs are the exactly same.  This means that the graph  is reflected over the x-axis. The axis of symmetry is x-axis.

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