Quadratic functions

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

f(x) = ax² + bx + c,
f(x) = a(x − h)² + k.

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:

h = −b/(2a);
k = f(h) = c − b²/(4a).

(We may 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 (h, k) 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. (That is, the absolute minimum or maximum is k, and this absolute extremum occurs at h.) The parabola is symmetric, with a vertical line of symmetry whose equation is x = h. The initial value of the function is f(0) = c, so the vertical intercept, or y-intercept, on the graph is (0, c). The roots (or zeroes) of the function (which correspond to the horizontal intercepts, or x-intercepts, on the graph) are given by the quadratic formula:

r_± = h ± √(-k/a) = [−b ± √(b² − 4ac)]/(2a).

However, these will be imaginary numbers if b² − 4ac is negative, or equivalently if k has the same sign as a, in which case the graph has no horizontal intercepts. The domain of any quadratic function is the set of all real numbers, while the range is [k, ∞) if a > 0 or (−∞, k] if a < 0.

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

(h, k) ―the vertex;
(0, c) ―the vertical intercept;
(2h, c);
(h + 1, k + a);
(h − 1, k + a);
(r₋, 0) ―one horizontal intercept;
(r₊, 0) ―the other horizontal intercept.

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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This web page was written between 2010 and 2018 by Toby Bartels, last edited on 2018 August 13. Toby reserves no legal rights to it.

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