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Quadratic Functions

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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 .

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 . 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 .

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 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.