How can I calculate the vertex of a parabola?

1 Answer
Jul 21, 2014

If parabola is defined by a quadratic function #y=ax^2+bx+c#, it's vertex is a point of extrema (minimum for #a>0# and maximum for #a<0#).

Therefore, the easiest way to determine this point is to take the first derivative from a given quadratic function and equate it to zero. The solution of this equation gives an #x#-coordinate of a vertex. The #y#-coordinate can be obtained by substituting the just found #x#-coordinate into a given quadratic function.

The first derivative of a function #y=ax^2+bx+c# is #2ax+b#. Therefore, we have to solve the following equation:
#2ax+b=0#

Solution:
#x=-b/(2a)#

Substituting this into a given quadratic function to get #y#-coordinate:
#y=a(-b/(2a))^2+b(-b/(2a))+c=b^2/(4a)-b^2/(2a)+c=(4ac-b^2)/(4a)#

Therefore, the #(x,y)# coordinates of a vertex of this parabola are
#(-b/(2a),(4ac-b^2)/(4a))#