Question

Explain Why the graph of a quadratic equation has the shape or a parabola?

Answer 1

In a nutshell:
A parabola is curved instead of linear, in your case it is probably just facing up or down so I won't get into square roots for now.
So the quadratic equation that you probably have had to memorize (or will soon) is:

x=(-b[+or-]√(b²-4ac))/2a when you have an equation like ax²+bx+c=0
Now where does the curve shape come from? You see that little pesky plus or minus in the equation? That's because there are always 2 values (inputs) that will generate the same output. 
Example:
y=x²
(2)²=4
(-2)²=...4
So if you were to follow this pattern, and plot the points on a graph, you would end up with a curve. You end up with a curve because the slope is constantly increasing. 

And this is actually where you start the study of Calculus(!), which is all about measuring slopes (And a bunch of other stuff, but this is the easiest part to explain). Actually, in this case of y=x², the slope at any given point (funnily enough) is equal to 2 times your x-value. 

The point is, your line is curved because unlike a linear equation, the slope is changing (at a constant rate).