Parabola

A parabola has a standard equation of

------(1)
or
------(2)
where (h,k) is the coordinate of the vertex (labeled as V in the graph).

For the parabola of type (1), the parabola is either opening upwards (p > 0) or downwards (p < 0). This type of parabola has a focus at F(h, k + p) and a directrix at y = k - p.


On the other hand, for the parabola of type (2), the parabola is either opening rightwards (p > 0) or leftwards (p < 0). This type of parabola has a focus at F(h + p, k) and a directrix at x = h - p.

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Hyperbola

A hyperbola which is centered at (h,k) has a standard equation

------(1)
if the hyperbola is opening sidewards, else, if it's opeing upwards and downwards, the standard equation will be
------(2)

Let's take a look at equation type (1):


This hyperbola in blue has
a) vertices at B1 and B2 at ,
b) foci at F1 and F2 at where ,
c) asymptotes in red dotted lines .


Now, let's look at equation type (2):


This hyperbola in blue has
a) vertices at B1 and B2 at ,
b) foci at F1 and F2 at where ,
c) asymptotes in red dotted lines .

Betty, Created with GeoGebra

Ellipse

An ellipse has a standard equation of



where
(h,k) is the center of the ellipse,
a is the distance between the vertex and the center, and
b is the distance between the endpoint of minor axis and the center.

Look at the graph below, point A is the center.
When a > b, points B1 and B2 are the vertices, points C1 and C2 are the endpoints of minor axis, and F1 and F2 are the foci.
When a < b, points C1 and C2 are the vertices, points B1 and B2 are the endpoints of minor axis, and F3 and F4 are the foci.
When a = b, a circle is formed. As such, we can say that circle is a special type of ellipse.

The line segment which connects the vertices is called as major axis. The line segment which is shorter, perpendicular to the major axis and passes through the center is called the minor axis.

You may slide the values of a, b, h or k to observe the changing of the graph.



Betty, Created with GeoGebra