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A quadratic function is a second degree polynomial function. The
general form of a quadratic function is this:
*f* (*x*) = *ax*
^{2} + *bx* + *c*
, where
*a*
,
*b*
, and
*c*
are real numbers, and
*a*≠ 0
.

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

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.

Figure %: The graph of the parabola
*y* = *a*(*x* - *h*)^{2} + *k*
. It is a quadratic
function in standard form. On the left
*a* < 0
, and on the right
*a* > 0
.

As was mentioned previously, one of the most important techniques to know is how to solve for the roots of a polynomial. There are many different methods for solving for the roots of a quadratic function. In this text we'll discuss three.

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)
.

Not all quadratic functions can be easily factored. Another method, called
completing the square, makes it easier to factor a quadratic function. When
*a* = 1
, a quadratic function
*f* (*x*) = *x*
^{2} + *bx* + *c* = 0
can be rewritten
*x*
^{2} + *bx* = *c*
. Then, by adding
()^{2}
to both sides, the left side can be
factored and rewritten
(*x* + )^{2}
. Taking the square root of both
sides and subtracting
from both sides solves for the roots.

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