In the applet below, move the sliders on the right to change the values of a, h and k and note the effects it has on the graph.
See also General Function Explorer where you can graph up to three functions of your choice simultaneously using sliders for independent variables as above. See also Linear Explorer and Cubic Explorer.
This form of a quadratic is useful when graphing because the vertex location is given directly by the values of h and k. In the graph above, click 'zero' under h and k, and note how the vertex is now at 0,0. The value of k is the vertical (y) location of the vertex and h the horizontal (x-axis) value. Move the sliders for h and k noting how they determine the location of the curve but not its shape.
The value of a is the same as with the standard form - it determines the 'steepness' of the parabola and the sign of a determines if the curve open upwards or downwards. Positive values open at the top. Adjust the slider for a and see this for yourself. Be sure to try both positive and negative values.
In the figure above, click on 'show roots'. As you play with the quadratic, note that the roots are where the curve crosses the x axis, where y=0. Usually there are two roots since the curve crosses the x-axis twice, so there are two different values of x where y=0.
If you make k zero, you will see that both roots are in the same place. Under some conditions the curve never crosses the x-axis and so the equation has no real roots.
When expressed in vertex form, the roots of the quadratic are given by the formula below. It gives the location on the x-axis of the two roots. If the expression inside the square root is negative, there are no real roots.
Click on "show axis of symmetry". This is a vertical line through the vertex of the curve. Note how the curve is a mirror image on the left and right of the line. (We say the curve is symmetrical about this line). Note too that the roots are equally spaced on each side of it.
When expressed in vertex form, the axis of symmetry of a quadratic is located at x=h.