For a quadratic function y=ax^2+bx+c, a maximum is there if a<0 and it has a minimum, if a>0. Please see below for details.
We can write y=ax^2+bx+c as
y=a(x^2+b/ax)+c
= a(x^2+2xxb/(2a)xx x+(b/(2a))^2-(b/(2a))^2)+c
= a(x^2+2xxb/(2a)xx x+(b/(2a))^2)-a(b/(2a))^2+c
= a(x+b/(2a))^2-b^2/(4a)+c
= a(x+b/(2a))^2-(b^2-4ac)/(4a)
Observe that as (x+b/(2a))^2 is always greater than 0,
if a is positive, we will have a minima for y, when x+b/(2a)=0 i.e. x=-b/(2a), which will be at -(b^2-4ac)/(4a), and
if a is negative, we will have a maxima for y, when x+b/(2a)=0 i.e. x=-b/(2a), which will be at -(b^2-4ac)/(4a).
Hence to find a maxima or minima for a quadratic function, observe the sign of a and convert the equation, as above, in form a(x-h)^2+k. Then the corresponding maxima or minima will be k, when x=h.
