By Nami Youn




 
 

Write-up  #2


Exploring the graph of 
 
 


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.



1. Varying the values of "a", when a is positive

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


(purple), (blue),(green),(yellow)
 






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.
 
 



2. Varying the values of "a", when a is negative


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

(purple),(blue), (green), (yellow).
 
 







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.



3. Relation between the graph  a>0 and a<0

Let's compare of the following graphs.
 
 

(purple), (green)
 
 



 
 
 
 
 
 

-4 -1 0 1 4
16 2 0 1 16
-16 -2 0 -16 -16

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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