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 may assume that a ≠ 0,
because otherwise our quadratic function is simply a linear function,
which we already know how to handle.)
- h = −b/(2a);
- k = f(h) =
c − ah2 =
c − b2/(4a).
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.
(That is, the absolute minimum or maximum is k,
and this absolute extremum occurs at h.)
The parabola is symmetric,
with a vertical line of symmetry
whose equation is x = h.
The initial value of the function is f(0) = c,
so the vertical intercept, or y-intercept, on the graph
is (0, c).
The roots (or zeroes) of the function
(which correspond to
the horizontal intercepts, or x-intercepts,
on the graph)
are given by the quadratic formula:
However, these will be imaginary numbers
if b2 − 4ac is negative,
or equivalently if k has the same sign as a,
in which case the graph has no horizontal intercepts.
The domain of any quadratic function is the set of all real numbers,
while the range
is [k, ∞) if a > 0
or (−∞, k] if a < 0.
- 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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