Quadratic Functions: Vertex Form
- Whit Ford
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. 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: - the vertex lies to the right, or left, of the y-axis - the vertex lies above, or below, the x-axis - the graph becomes a horizontal line, or opens down - any part of the graph passes through the other blue point on the graph (-3, -1) - the vertex of the graph (the blue point labelled V) is moved on top of the other blue point on the graph: (-3, -1) 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. h determines the x-coordinate of the graph's vertex. Note that in the equation shown on the graph, when x is equal to h, the value in parentheses must equal zero, which is the smallest value that any squared real quantity can assume. Therefore, when a is positive, h becomes the x-coordinate at which the graph must reach its lowest point: its vertex. For all values of x other than h, the squared quantity in parentheses must produce a value greater than zero (higher than the vertex). k determines the y-coordinate of the graph's vertex. When (at the vertex), the entire squared term will always equal zero, and the result of the equation must equal k. This forces the y-coordinate of the vertex to become k. These three values, a, h, and k, 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. 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/