Standard Form of the equation of a parabola is used often, as it is what you end up with after multiplying two binomial factors together, then simplifying.
The coefficients of each term in Standard Form, a, b, and c, are required when using the Quadratic Formula to find the x-intercepts of the graph of a parabola.
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.

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, or opens down
- some part of the graph passes through the blue point on the graph: (-3, -1)
- the vertex of the graph (the blue point labelled V) passes through the blue point on the graph: (-3, -1). This is much more challenging!
a is referred to as the "dilation factor". It determines how much the graph is stretched away from, or compressed towards, the x-axis. Note what happens to the graph when you set a to a negative value.
c shifts (translates) the graph vertically.
b alters the the graph in a complex way. How would you describe the effect that changing the value of b has on the graph? If you wish to explore this behavior in a bit more depth, you may use this applet: http://tube.geogebra.org/material/simple/id/648429.
These three values, a, b, and c, 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 that may be a bit more intuitive.
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/