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 ([i]x[/i]-intercepts).[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.[br][br]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:[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[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 purple point labelled V) passes through the blue point on the graph: (-3, -1). This is more challenging![br][br][b]a[/b] is referred to as the "dilation factor". It either stretches the parabola away from the [i]x[/i]-axis, or compresses it 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]M[/b] and [b]N[/b] are referred to as the "roots" or the "zeroes" of the function. They determine where the function will cross the [i]x[/i]-axis.[br][br]These three values, [b]a[/b], [b]M[/b], and [b]N[/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.[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]

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]

The Vertex Form of the equation of a parabola is very useful. It is helpful when analyzing a quadratic equation, and it can also be helpful when creating an equation that fits some data.[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 understand 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, or below, the [i]x[/i]-axis[br][br]- the graph becomes a horizontal line, or opens down[br][br]- any part of the graph passes through the other blue point on the graph (-3, -1)[br][br]- the vertex of the graph (the blue point labelled V) is moved on top of the other blue point on the graph: (-3, -1)[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]h[/b] determines the [i]x[/i]-coordinate of the graph's vertex. Note that in the equation shown on the graph, when [i]x[/i] is equal to [b]h[/b], the value in parentheses must equal zero, which is the smallest value that any squared real quantity can assume. Therefore, when [b]a[/b] is positive, [b]h[/b] becomes the [i]x[/i]-coordinate at which the graph must reach its lowest point: its vertex. For all values of [i]x[/i] other than [b]h[/b], the squared quantity in parentheses must produce a value greater than zero (higher than the vertex).[br][br][b]k[/b] determines the [i]y[/i]-coordinate of the graph's vertex. When [math]x = h[/math] (at the vertex), the entire squared term will always equal zero, and the result of the equation must equal [b]k[/b]. This forces the [i]y[/i]-coordinate of the vertex to become [b]k[/b].[br][br]These three values, [b]a[/b], [b]h[/b], and [b]k[/b], will describe a unique parabola. To completely describe any parabola, all someone needs to know is: its dilation factor and the coordinates of its vertex. There are also other ways of describing everything about a parabola, but this is often one of the simplest ways of doing so.[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]