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 − ah² = 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. This vertex is the only turning point of the graph; the graph turns upwards at the vertex is if a > 0, and it turns downwards if a < 0. 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. The parabola is symmetric, with a vertical line of symmetry whose equation is x = h. Since f(0) = c, the vertical intercept, or y-intercept, on the graph is (0, c). The horizontal intercepts, or x-intercepts, are (r₊, 0) and (r₋, 0), where r₊ and r₋ are given by the quadratic formula:

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

(It doesn't really matter which one is r₊ and which one is r₋.) 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.

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 by Toby Bartels, last edited on 2019 December 3. Toby reserves no legal rights to it.

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