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).

Assuming that a ≠ 0, 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 is the lowest point on the parabola; if a < 0, then the vertex is the highest point on the parabola. The parabola is symmetric, with a vertical line of symmetry whose equation is x = h. The zeroes of the function are given by the quadratic formula:

z_± = [−b ± √(b² − 4ac)]/2a.

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

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);
(z₋, 0) (one horizontal intercept);
(z₊, 0) (the other horizontal intercept).

The last two won't exist on the graph if z_± are imaginary.

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This web page was written in 2010 by Toby Bartels, last edited on 2010 November 28. Toby reserves no legal rights to it.

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