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.graph of the parabola

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).