page 1 of 2

Page 1

Page 2

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.

Page 1

Page 2

Take a Study Break!