The functions y=x²

$y=-x^2$

The most basic quadratic function $a=-1$,$b=0$,$c=0$
Maximum, sad face function, its vertex is $(0,0)$
The axis of symmetry of this function is $X=0$.
The interval of increase of the function: $X<0$
The interval of decrease of the function: $X>0$
Set of positivity: None. The entire parabola is below the axis$X$.
Set of negativity: All $X$ except for $X=0$

$y=ax^2$

Properties of the function:
The quadratic function 
$any number = a$,$b=0$,$c=0$

Its vertex is $(0,0)$
The axis of symmetry of this function is $X=0$.

As $a$  increases, the parabola will have a smaller opening - closer to its axis of symmetry.
As $a$  decreases, the parabola will have a larger opening - further from its axis of symmetry.