*f*(*x*) =*a**x*^{2}+*b**x*+*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*/(2*a*);*k*=*f*(*h*).

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.
(Technically, the absolute minimum or maximum *is* *k*,
and 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**:

*r*_{±}= [−*b*± √(*b*^{2}− 4*a**c*)]/(2*a*).

In general, there are up to 7 useful points on the graph:

- (
*h*,*k*) ―the vertex; - (0,
*c*) ―the vertical intercept; - (2
*h*,*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 between 2010 and 2016 by Toby Bartels, last edited on 2016 August 10. Toby reserves no legal rights to it.

The permanent URI of this web page
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