# How to find the Vertex of a Parabola

A parabola is a curve which is given by a quadratic equation: *y = ax ^{2 }+ bx + c*

Every parabola contains a vertex. A vertex is the maximum or minimum point on the curve, at which the gradient of the curve is zero. This article is a tutorial in how to find the coordinates of this vertex

## How to Find the Vertex Coordinates

1. Write the parabola equation in the form: *y = ax ^{2} + bx + c* (where

*a*,

*b*and

*c*are constants)

2. Find the gradient of the parabola by differentiating the parabola equation:

*y = ax ^{2} + bx + c* differentiates to give

*dy/dx = 2ax + b*

3. Find the value of x which gives a gradient of zero:

*dy/dx = 2ax +b = 0* when *x = -b / 2a*

4. Find the value of y which corresponds to this x, by putting *x = -b / 2a* back into the parabola equation:

*y = a (-b / 2a) ^{2} + b (-b / 2a) + c*

*= b ^{2 }/ 4a - b^{2 }/ 2a + c*

*= -b ^{2 }/ 4a +c*

5. The co-ordinates of the vertex are therefore *(-b / 2a, -b ^{2 }/ 4a + c)*

## Example

An example parabola is shown in the diagram.

1. The equation of this parabola is *y = -2x ^{2} - 4x + 6*

2. The gradient is *dy/dx = -4x - 4*

3. The gradient is zero when *x = -1*

4. When *x = -1, y = -2(-1) ^{2} - 4(-1) + 6 = -2 + 4 + 6 = 8*

5. The vertex coordinates are therefore *(-1, 8)*

## Mathematics Textbooks

This is the book I used throughout my degree. It provides a great grounding in college-level mathematics for anyone studying math or a physical science.

## Maximum or Minimum?

One last question remains to be answered. How do we know whether the vertex we have found from the equation is a maximum or a minimum point?

There is a simple test, which can be performed as follows:

1. Take the equation for the gradient, *dy/dx, *and differentiate it again by x, to get* d ^{2}y/dx^{2.}*

* ^{}*2. This quantity

*d*

^{2}

*y/dx*

*2*is the rate of change of the gradient.

If *d*^{2}*y/dx**2 **<0, *then the gradient is decreasing as x increases. This means that the gradient of the parabola is positive on the left of the vertex and negative on the right. The vertex is a maximum.

If *d*^{2}*y/dx**2 **>**0, *then the gradient is increasing as x increases. This means that the gradient of the parabola is negative on the left of the vertex and positive on the right. The vertex is a minimum.

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## Comments 3 comments

Interesting. This takes me back to my old school days, when I actually understood and enjoyed math/calculus.

Cheers, Louise :)

If only I had found such a helpful guide when I was still in high school! GREAT Hub!

Another very clearly explained topic. Might I be so bold as to suggest you might add some applications of this Mathematics. Voted up!