The graph of a

**quadratic** function is called
a

parabola. A parabola is roughly shaped like the letter "U" -- sometimes it is just this way, and other times it is upside-down. There
is an easy way to tell whether the graph of a

**quadratic** function
opens upward or downward: if the

leading coefficient is greater than zero, the parabola opens upward, and if the leading coefficient is less than zero, the parabola opens downward.
Study the graphs below:

*Figure 2.1: On the left, **y* = *x*^{2}. On the right, *y* = - *x*^{2}.

The function above on the left,

*y*
= *x*^{2}, has leading coefficient

*a* = 1≥ 0, so the parabola opens
upward. The other function above, on the right, has leading coefficient

-1, so the parabola opens
downward.

The

standard form of a

**quadratic** function is a little different from the general
form. The standard form makes it easier to graph. Standard form looks like this:

*f* (*x*) = *a*(*x*
- *h*)^{2} + *k*, where

*a*≠ 0. In standard form,

*h*
= - and

*k* = *c* - . The point

(*h*, *k*) is called the

vertex of the parabola. The line

*x* = *h* is called the

axis of the parabola. A parabola is symmetrical with respect to its axis. The value of the function at

*h*
= *k*. If

*a* < 0, then

*k* is the maximum value
of the function. If

*a* > 0, then

*k* is the minimum value of
the function. Below these ideas are illustrated.

Factoring

Factoring is a technique taught in algebra, but it is useful to review here. A **quadratic** function has three terms. By setting the function equal to zero and factoring
these three terms a **quadratic** function can be expressed by a single
term, and the roots are easy to find. For example, by factoring the **quadratic**
function *f* (*x*) = *x*^{2} - *x* - 30, you get *f*
(*x*) = (*x* + 5)(*x* - 6). The roots of *f* are *x*
= { -5, 6}. These are the two values of *x* that make the function *f*
equal to zero. You can check by graphing the function and noting in which two places the graph intercepts the *x*-axis.
It does so at the points (- 5, 0) and (6, 0).