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.
This vertex is the only turning point of the graph;
the graph turns upwards at the vertex is if a > 0,
and it turns downwards if a < 0.
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.
The parabola is symmetric,
with a vertical line of symmetry
whose equation is x = h.
Since f(0) = c,
the vertical intercept, or y-intercept, on the graph
is (0, c).
The horizontal intercepts, or x-intercepts,
are (r+, 0)
and (r−, 0),
where r+ and r−
are given by the quadratic formula:
(It doesn't really matter
which one is r+ and which one is r−.)
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.
- 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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This web page was written by Toby Bartels, last edited on 2019 December 3.
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