*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*).

Assuming that *a* ≠ 0,
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 parabola.
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*± √(*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 in 2010 and 2011 by Toby Bartels, last edited on 2011 November 7. Toby reserves no legal rights to it.

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