# Calculus: How to Graph an Ellipse

An ellipse is basically a circle that has been squished either horizontally or vertically. From a pre-calculus perspective, an *ellipse* is a set of points on a plane, creating an oval, curved shape such that the sum of the distances from any point on the curve to two fixed points (the *foci*) is a constant (always the same).

Graphically speaking, you must know two different types of ellipses: horizontal and vertical. A horizontal ellipse is short and fat; a vertical one is tall and skinny. Each type of ellipse has these main parts:

**Center.**The point in the middle of the ellipse is called the*center*and is named (*h, v*) just like the vertex of a parabola and the center of a circle.**Major axis.**The*major axis*is the line that runs through the center of the ellipse the long way. The variable*a*is the letter used to name the distance from the center to the ellipse on the major axis. The endpoints of the major axis are on the ellipse and are called*vertices.***Minor axis.**The*minor axis*is perpendicular to the major axis and runs through the center the short way. The variable*b*is the letter used to name the distance to the ellipse from the center on the minor axis. Because the major axis is always longer than the minor one,*a*>*b.*The endpoints on the minor axis are called*co-vertices.***Foci.**The*foci*are the two points that dictate how fat or how skinny the ellipse is. They are always located on the major axis, and can be found by the following equation:*a*^{2}–*b*^{2}=*F*^{2}where*a*and*b*are mentioned as in the preceding bullets and*F*is the distance from the center to each focus.The labels of a horizontal ellipse and a vertical ellipse.

Figure a shows a horizontal ellipse with its parts labeled; Figure b shows a vertical one. Notice that the length of the major axis is 2*a* and the length of the minor axis is 2*b.** *This figure also shows the correct placement of the foci — always on the major axis.

Two types of equations apply to ellipses, depending on whether they're horizontal or vertical. The horizontal equation has the center at (*h, v*), major axis of 2*a, *and minor axis of 2*b:*

The vertical equation has the same parts, although *a *and *b *switch places:

Note that when the bigger number *a* is under *x,* the ellipse is horizontal; when the bigger number is under *y,* it's vertical.