**Mathematics
Education**

**EMAT 6680,
Professor Wilson**

**
Exploration 2, Parabolas by Ursula Kirk**

Fix the values for ** a** and

**
**

·
Try an animation for
the same range

·
What is happening
mathematicatically?

·
Can you prove this is
a translation and that the shape of the parabola does not change?

**
**

,

** What are the mathematics? **

The graphs have all the
same shape with no translations over the x-axis. However, by changing c, we
obtain graphs with translations over the y-axis. As we move down from c=5 to c -5.
The vertices of our graphs have shifted 5/4 units down. When c is positive, the graphs intercept the
y-axis at positive values. When c is zero, the graph intercepts the y-axis at
zero. When c is negative, the graphs intercept the y-axis at negative values.

The graphs have been
shifted to the left by a unit of 1, since b=1. All the graphs are concave up,
since a>0.

**
Vertex Form**

Now we
will convert our first equation from standard form to vertex form by completing
the square.

Therefore
for this first equation our vertex is

The
rest of the parabolas will have the same h, but k will change by shifting down
by 5/4.

**Number
of Solutions**

Not
all parabolas have solutions, when ,
the parabolas do not have any real solutions and their solutions are imaginary,
when .
The parabolas have two real solutions. At the parabola will have one single real
solution.

**Animation**

,