Quadratic Functions
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 .
Graphing Quadratic
Functions
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 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 . Ifa <
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 .
Solving Quadratic
Equations
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
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) .
Completing the Square
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.
The Quadratic Equation
For quadratic functions that can't be solved
using either of the previous two methods, the quadratic equation can be used.
If f (x) = ax 2 + bx + c = 0 , then the quadratic equation states
that x =
.
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