*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.
This vertex is the only *turning point* of the graph;
the graph turns upwards at the vertex 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**:

*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.

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This web page was written by Toby Bartels, last edited on 2024 August 15. Toby reserves no legal rights to it.

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