How to use
Click the "+" and "-" buttons
under each value to change that value.
Holding a button down causes the action to be repeated.
Examples
Other Notes
The values of h and k give
horizontal and vertical (resp.) translation distances,
and t gives rotation angle (measured in degrees).
Notice how changes in these transformation values affect the coefficients,
and how changes in the coefficients affect the transformations.
The lines shown in green in the graph are
the following key lines for the conic sections: the major and minor
axes for ellipses (crossing at the center of the ellipse),
the axis of symmetry and perpendicular line
through the vertex for a parabola (crossing at the vertex),
and the two perpendicular axes of symmetry (crossing through
the center point) for a hyperbola. In all cases, the two lines cross
at the point (h,k), and are rotated from the position
parallel to the coordinate axes by t degrees.
In graphs of hyperbolas, the asymptotes of the hyperbola are shown
as orange lines.
The "Type:" label displays what type of conic
section is shown in the graph. This can be determined by the
value of the discriminant B^{2}-4AC:
- If B^{2}-4AC>0, then the graph is a hyperbola.
- If B^{2}-4AC=0, then the graph is a parabola.
- If B^{2}-4AC<0, then the graph is an ellipse
(if B=0 and A=C in this case, then the graph
is a circle).
One other important formula determines the relationship between the
coefficients and the angle of rotation:
tan(2t)=B/(A-C).
Note that rotation has no effect on the values of
the coefficients D, E, and F, and
that t=0 (no rotation) if and only if B=0.
The values of the coordinates of the point (h,k)
are best determined from the coefficients
by first reversing the effect of the rotation
(so that B=0), then completing the squares.