Quadratic functions

A quadratic function f may be written in either of two forms:

You can move from the second form to the first by expanding; you can move from the first to the second by completing the square or by using these formulas:

(We assume that a ≠ 0, because otherwise our quadratic function is simply a linear function, which we already know how to handle.)

If (as we assume) it's not linear, then the graph of a quadratic function is a shape called a parabola. The point (hk) on the graph is called the vertex of the parabola. If a > 0, then the vertex gives the absolute minimum of the function; if a < 0, then the vertex gives the absolute maximum of the function. The parabola is symmetric, with a vertical line of symmetry whose equation is x = h. The roots (or zeroes) of the function are given by the quadratic formula:

However, these will be imaginary numbers if b2 − 4ac is negative.

In general, there are up to 7 useful points on the graph:

Some of these points might happen to be the same as others, and the last two won't exist on the graph if the roots r± are imaginary. However, there are always at least three distinct real points on this list.
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