Factored Form of the equation of a parabola is used often, as it is often the easiest way to create the equation of a parabola with two specific roots (x-intercepts).
The graph below contains three green sliders. Click on the circle in a slider and drag it to the left or right, while watching the effect it has on the graph.
The graph will display the coordinates of the points where the graph intersects each axis, as well as the coordinates of the vertex of the parabola.

Once you have a feel for the effect that each slider has, see if you can adjust the sliders so that:
- the vertex lies to the right, or left, of the y-axis
- the vertex lies above the x-axis
- the graph becomes a horizontal line
- some part of the graph passes through the blue point on the graph: (-3, -1)
- the vertex of the graph (the purple point labelled V) passes through the blue point on the graph: (-3, -1). This is more challenging!
a is referred to as the "dilation factor". It either stretches the parabola away from the x-axis, or compresses it towards the x-axis. Note what happens to the graph when you set a to a negative value.
M and N are referred to as the "roots" or the "zeroes" of the function. They determine where the function will cross the x-axis.
These three values, a, M, and N, will describe a unique parabola. To completely describe any parabola, all someone needs to tell you are these three values. However, there are also other ways of describing everything about a parabola.
If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/