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 (hk) 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:

• (hk) (the vertex);
• (0, c) (the vertical intercept);
• (2hc);
• (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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