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 (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. 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:

(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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