*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*) =*c*−*a**h*^{2}=*c*−*b*^{2}/(4*a*).

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**:

*r*_{±}=*h*± √(-*k*/*a*) = [−*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.

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2019 April 29. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2022FA/quadratic/`

.