# Quadratic functions (§4.3)

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).
(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:

• r± = [−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);
• (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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