# Graph of the Quadratic Function

Quadratic function is function of the form y=ax^2+bx+c where a,b,c are arbitrary constants and a!=0.

To draw graph of this function we need to perform following transformations of function (also known as "completing the square"): ax^2+bx+c=a(x^2+b/a x)+c=a((x^2+2* b/(2a) x+(b^2)/(4a^2))-(b^2)/(4a^2))+c=a((a+b/(2a))^2-(b^2)/(4a^2))+c=

=a(x+b/(2a))^2-(b^2)/(4a)+c=a(x+b/(2a))^2+(4ac-b^2)/(4a).

So, color(red)(ax^2+bx+c=a(x+b/(2a))^2+(4ac-b^2)/(4a)).

So, to draw graph of the function y=ax^2+bx+c we need to do following steps:

1. Draw graph of the function y=x^2.
2. Stretch (or compress) graph with the coeffcients |a|. If a<0 reflect obtained graph about x-axis.
3. Move obtained graph (4ac-b^2)/(4a) units up if (4ac-b^2)/(4a)>0 and down otherwise.
4. Move obtained graph b/(2a) units left if b/(2a)>0 and right otherwise.

Note, that these steps can be performed in any order.

Line x=-b/(2a) is called axis of symmetry of parabola, that is graph of the function y=ax^2+bx+c. Point at which parabola intersects axis of symmetry is called vertex of parabola.

If a>0, then parabola is open upward. In this case function is decreasing on (-oo,-b/(2a)) and increasing on (-b/(2a),+oo). If a<0, then parabola is open downward. In this case function is increasing on (-oo,-b/(2a)) anddecreasing on (-b/(2a),+oo) (see figure).