Three sliders allow you to change the coefficients of a quadratic equation from -5 to 5.[br]The resulting parabola from the equation [math]y = a x^2 + b x + c [/math] is shown with important points of [br][list=1][br][*] The Vertex[br][/*][*] y-Intercept[br][/*][*] Roots [math] x_1 [/math] and [math] x_2 [/math] [br][/*][*] The line of symmetry[br][/*][/list]Also, shown is the [math]h,k[/math] form of the quadratic equation [math]y=a\left(x-h\right)^2+k[/math] where [math]\left(h,k\right)[/math] are the coordinates of the vertex. [br][br]You can turn on tracing of the vertex to see how the vertex moves as you vary the quadratic coefficients.

Explorations:[br] What changes when [math]c[/math] is varied?[br] What is different if [math]a[/math] < 0 or [math]a[/math] > 0 ?[br] How does the shape change with larger [math]a[/math] values?[br] How does [math]b[/math] move the graph?[br] Under what conditions do you get x-Intercept (Roots, [math]x_1 , x_2[/math]) values?[br] Set [math]b=0[/math], then how does [math]a[/math] vary the curve?[br] Try to come up with equations for the path (trace) of the vertex as you vary each coefficient.

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.[br][br]Notice how the distance from the Vertex to the focus changes as the parabola gets shallower and steeper.