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.[br][br]The coefficients of each term in Standard Form, [b]a[/b], [b]b[/b], and [b]c[/b], are required when using the Quadratic Formula to find the [i]x[/i]-intercepts of the graph of a parabola.[br][br]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:[br][br]- the vertex lies to the right, or left, of the [i]y[/i]-axis[br][br]- the vertex lies above the [i]x[/i]-axis[br][br]- the graph becomes a horizontal line, or opens down[br][br]- some part of the graph passes through the blue point on the graph: (-3, -1)[br][br]- 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![br][br][b]a[/b] is referred to as the "dilation factor". It determines how much the graph is stretched away from, or compressed towards, the [i]x[/i]-axis. Note what happens to the graph when you set [b]a[/b] to a negative value.[br][br][b]c[/b] shifts (translates) the graph vertically.[br][br][b]b[/b] alters the the graph in a complex way. How would you describe the effect that changing the value of [b]b[/b] has on the graph? If you wish to explore this behavior in a bit more depth, you may use this applet: [url=http://tube.geogebra.org/material/simple/id/648429]http://tube.geogebra.org/material/simple/id/648429[/url].[br][br]These three values, [b]a[/b], [b]b[/b], and [b]c[/b], 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.[br][br]If you wish to use other applets similar to this, you may find an index of all my applets here: [url=https://mathmaine.wordpress.com/2010/04/27/geogebra/]https://mathmaine.com/2010/04/27/geogebra/[/url]