Vertex of a parabola

Vertex of upside down parabola

A parabola is a graph of a quadratic equation/function. It is shaped in the form of an upright or an upside down 'U'.

The vertex of a parabola is the topmost or the bottom-most point of the 'U' curve according to whether the U curve is upside down or upside up. In order to graph a parabola, you need to get the coordinates of the vertex. The coordinates of the vertex of a parabola can be obtained from the quadratic equation/function of that parabola.

Vertex of U shaped parabola

There are different methods of computing the coordinates of the vertex from a quadratic equation/function, depending on the form in which it is written. For example, if the quadratic equation is given in the general form, then the method of calculating its vertex is different from when it is given in the vertex form.

Let us study the method of calculating the vertex with the help of examples:

A) When the quadratic equation/function is in the general form

When the given quadratic equation/function is in the general form, then you need to get the x and y coordinates of the vertex by the following method:
For any quadratic equation  

ax^2 + bx + c = 0,

the x coordinate of its vertex is given by:

x = -b/a

For example, in the quadratic function

f(x) = 3x^2 + 4x + 5,

the x coordinate of its vertex is

x = -4/3

In order to get the y coordinate of the vertex, we need to substitute the value of the x-coordinate obtained above in place of 'x' in the quadratic. That is, we get

y = 3(-4/3)^2 + 4(-4/3) + 5
y = 5

Therefore the y-coordinate of the vertex is 5.
Therefore the coordinates of the vertex are  

Vertex = (-4/3, 5).

B) When the quadratic equation/function is given in the vertex form

When the given quadratic is in the vertex form, it is easier to get the coordinates of its vertex directly from the quadratic.

For any quadratic in the vertex form:

f(x) = a(x - h)^2  k,

Vertex is given by

Vertex = (h, k)

For example, in the quadratic:

f(x) = 3(x - 4)^2 + 5,

the vertex is

vertex = (4, 5)