# Section 9-3

# Quadratic Functions

The general form of a quadratic function is **f****(***x*) = *ax*^{2} + *bx* + *c*. The graph of such a function is a parabola, a type of 2-dimensional curve.

The "basic" parabola, **y**** = ***x*^{2}, looks like this:

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

If the coefficient of *x*^{2} is 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":

**f****(***x*) = *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:

**f****(***x*) = *a*(*x *– *h*)(*x *– *h*) + *k*

**f****(***x*) = *a**x*^{2} –** 2***a**hx* + *a**h*^{2} + *k*

The coefficient of *x* here is –2*a**h*. This means that in the standard form, *f*(*x*) = *ax*^{2} + *bx* + *c*, the expression

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

Example:

Find the vertex of the parabola.

**f****(***x*) = 3*x*^{2} + 12*x* –** 12 **

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

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

**f****(**–**2) = 3(**–**2)**^{2} + 12(–**2) **–** 12**

**= **–**24 **

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

## AXIS OF SYMMETRY

The axis of symmetry of a parabola is the vertical line through the vertex. For a parabola in standard form, **f****(***x*) = *ax*^{2} + *bx* + *c*, the axis of symmetry has the equation

Example:

Find the axis of symmetry.

**f****(***x*) = 2*x*^{2} + *x* –** 1**

Here, *a *= 2 and *b* = 1. So, the axis of symmetry is the vertical line

## DOMAIN AND RANGE

As with any function, the domain** **of a quadratic function *f*(*x*) is the set of *x*-values for which the function is defined, and the range is the set of all the output values (values of *f*).

Parabolas generally have the whole real line as their domain: any *x* is a legitimate input. The range is restricted to those points greater than or equal to the *y*-coordinate of the vertex (or less than or equal to, depending on whether the parabola opens up or down).