Polynomial Functions: 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) = ax² + 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 = x². On the right, y = - x².

The function above on the left, y = x², 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

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) = x² - 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) = x² + bx + c = 0 can be rewritten x² + 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) = ax² + bx + c = 0, then the quadratic equation states that x = .