A quadratic function is any function that can be written in the form , where a, p, and q and . This is called the standard form, or the vertex form of the equation of a quadratic function. This form is the combination of three transformations.
* The effect of changing q in .
Changing q results in a vertical translation of a parabola on the xy-plane. The graph moves up if q is positive, and moves down if q is negative.
Example: The graph of moves up 4 units to create a new graph of the equation.

Or the original moves down 5 units by subtracting 5, turning into

* The effect of changing p in
Changing p results in a horizontal translation of a parabola on the xy-plane. The graph moves left if p is negative, and moves right if p is positive.
Example: The graph of is the image of after a horizontal translation of -7.

Or the graph of is the image of after a horizontal translation of +6.

* The effect of changing a in
Changing a results in a size change and/or the parabola’s open direction.
– The graph is stretched when |a|>1, and is compressed when 0<|a| <1. Example: The graph of [latex]y=4x^2[/latex] is thinner than the graph of [latex]y=x^2[/latex]
The graph of opens wider than the graph of .

– The graph opens up when a is positive and opens down when a is negative.

*This form is called the vertex form because the vertex is indicated: (p,q)
Example: has the vertex(-3,6)