Rayen Antillanca.
Assignment 2
Fix
the values for a and b, vary c. Make at least 5 graphs on the
same axes as you vary c. For the general function of the parabola,
namely:
Then if a=1 and b=1, c takes values
from 4 to 4, the equation is 

Where does
the parabola cut the yaxe? 

As
you can see, the parabola keeps the shape but it moves over the yaxe. The
value of b=1 indicates that the vertex of the parabola has a translation to
the left of the origin. Also, it means, the parabola makes a translation
depend on the value of c. In this case, as c is positive, the cut of the
parabola in the yaxe is in the positive side, and when c is negative, the
cut happen in the negative side of the yaxe. 
Now, we will fix a=1, b=1 and c
varies from 4 to 4. 

Where does
the parabola cut the yaxe? 

As
you can see, the parabola keeps the shape but it moves over the yaxe. The
value of b=1 indicates that the vertex of the parabola has a translation to
the right of the origin. Also, it means, the parabola makes a translation
depend on the value of c. In this case, as c is positive, the cut of the
parabola in the yaxe is in the positive side, and when c is negative, the
cut happen in the negative side of the yaxe. 
Summary 
C is the value where the parabola cuts the yaxe. When c is
a negative number, the parabola cuts in that negative number in the yaxe.
When c is a positive number the parabola cuts in that positive number in the
yaxe. When c=0 the parabola cuts in zero the yaxe of Cartesian plane. 
Now you can see the animation 

As
you can see in the prior animation, when the c varies from 4 to 4 the
parabola moves through the yaxis. This implies a translation of the
parabola. Recall, a translation means to move every point of the parabola, in
this case a constant distance in a specific direction. 
Summary 
Consider the equation If I complete square I get then the
vertex of this parabola is moved from the
origin to the right, and it is moved upward from the
origin. As you know, when c=0 the parabola cuts the yaxe in the origin. Then, if we vary c, we can see
that this function is the same parabola, but the parabola is moved upward or
downward depend of the value of c. 
Note: All graphs of this webpage were made with
Graphing Calculator 4.0 