How to find the vertex of a equation

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The vertex of a parabola is the point where the parabola crosses its axis of symmetry.   If the coefficient of the x 2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “ U ”-shape.  If the coefficient of the x 2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “ U ”-shape.

The standard equation of a parabola is

y = a x 2 + b x + c .

But the equation for a parabola can also be written in "vertex form":

y = a ( x − h ) 2 + k

In this equation, the vertex of the parabola is the point ( h , k ) .

You can see how this relates to the standard equation by multiplying it out:

y = a ( x − h ) ( x − h ) + k y = a x 2 − 2 a h x + a h 2 + k .

This means that in the standard form, y = a x 2 + b x + c , the expression − b 2 a gives the x -coordinate of the vertex.

Example:

Find the vertex of the parabola.

y = 3 x 2 + 12 x − 12

Here, a = 3 and b = 12 . So, the x -coordinate of the vertex is:

− 12 2 ( 3 ) = − 2

Substituting in the original equation to get the y -coordinate, we get:

y = 3 ( − 2 ) 2 + 12 ( − 2 ) − 12 = − 24

So, the vertex of the parabola is at ( − 2 , − 24 ) .