Three sliders allow you to change the coefficients of a quadratic equation from -5 to 5. The resulting parabola from the equation is shown with important points of
1. The Vertex
2. y-Intercept
3. Roots and
4. The line of symmetry
Also, shown is the form of the quadratic equation where are the coordinates of the vertex. You can turn on tracing of the vertex to see how the vertex moves as you vary the quadratic coefficients.
Explorations: What changes when is varied? What is different if < 0 or > 0 ? How does the shape change with larger values? How does move the graph? Under what conditions do you get x-Intercept (Roots, ) values? Set , then how does vary the curve? Try to come up with equations for the path (trace) of the vertex as you vary each coefficient.

## Movable Vertex Description

Below is an illustration where you can move the Vertex and y intercept. The quadratic equations are shown as well as two other properties of a parabola, the Directrix and the focus. One definition of a parabola uses a line segment perpendicular to the Directrix from the Directrix to the quadratic curve intersection point and the line segment from the intersection point to the focus. For a parabola the lengths of both line segments would be equal originating from any point on the Directrix. General parabolas that are not quadratic functions can be defined from a Directrix line and a focus point. Notice how the distance from the Vertex to the focus changes as the parabola gets shallower and steeper.