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:
(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:
However, these will be imaginary numbers
if b2 − 4ac is negative.
- r± =
In general, there are up to 7 useful points on the graph:
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.
- (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.
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last edited on 2012 September 30.
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