Quadratic Function

The general form of a quadratic function is f(x)=ax2+bx+c 

The graph of a quadratic function is a parabola , a type of 2 -dimensional curve.

The "basic" parabola, y=x2 looks like this:

The function of the coefficient a$a$ in the general equation is to make the parabola "wider" or "skinnier", or to turn it upside down (if negative)

f the coefficient of x2is positive, the parabola opens up; otherwise it opens down.

The Vertex

The vertex of a parabola is the point at the bottom of the " U " shape (or the top, if the parabola opens downward).

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)

The coefficient of x here is −2ah. This means that in the standard form, y=ax2+bx+c , the expression

−b2a

gives the x$x$ -coordinate of the vertex.

Example:

Find the vertex of the parabola.

y=3x2+12x−12

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

−122(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 parabola is (-2,-24)

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