Question

How can I calculate the vertex of a parabola?

Answer 1

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))`