Problem in graph of Quadratic Equation

[Neha Sanghvi]

Hey, could someone please help me with this problem? I just want to know why the graph of Quadratic Equation are symmetric and why does the maxima and minima ( = -b/2a ) the mean of the two roots? One more thing I didn't get was what does the a represent in ax2+bx+c ( ax square+bx+c)? I know for b and c, but couldn't figure out for a? Is it for the slope or whether the graph will be > 0 or < 0?

 

[Galileo]

If you write ax^2+bx+c in the more 'insightful' form:

a\left(x+\frac{b}{2a}\right)^2+\left(c-\frac{b^2}{4a}\right)

You can see that it is equal to the graph of ax^2, but shifted vertically by c-\frac{b^2}{4a} and horizontally by \frac{b}{2a}. The a can be seen as a scaling factor in the y-direction. If it's positive, you have a 'dale-parabola', if it's negative it's a 'hill-parabola'.