# How to Graph an Ellipse

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). An ellipse is basically a circle that has been squished either horizontally or vertically.

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:

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.

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* .

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* .

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:

where *a* and *b* are mentioned as in the preceding bullets, and *F* is the distance from the center to each focus.

This figure shows a horizontal ellipse and a vertical ellipse with their parts labeled. 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 is

with the center at (*h*. *v* ), major axis of 2*a*. and minor axis of 2*b* .

The vertical equation is

with the same parts — although *a* and *b* have switched places.

When the bigger number *a* is under *x*. the ellipse is horizontal; when the bigger number is under *y*. it’s vertical.

You have to be prepared to not only graph ellipses, but also to name all their parts. If a problem asks you to calculate the parts of an ellipse, you have

to be ready to deal with some ugly square roots and/or decimals. The following presents the parts for both horizontal and vertical ellipses.

**Horizontal Ellipse**

Vertices: (*h* ± *a*. *v* )

Co-vertices: (*h*. *v* ± *b* )

Length of Major Axis: 2*a*

Length of Minor Axis: 2*b*

**Vertical Ellipse**

Vertices: (*h*. *v* ± *a* )

Co-vertices: (*h* ± *b, v* )

Length of Major Axis: 2*a*

Length of Minor Axis: 2*b*

To find the vertices in a horizontal ellipse, use (*h* ± *a*. *v* ); to find the co-vertices, use (*h*. *v* ± *b* ). A vertical ellipse has vertices at (*h*. *v* ± *a* ) and co-vertices at (*h* ± *b*. *v* ).

For example, look at

which is already in the proper form to graph. You know that *h* = 5 and *v* = –1 (switching the signs inside the parentheses).

This example is a vertical ellipse because the bigger number is under *y*. so be sure to use the correct formula. This equation has vertices at (5, –1 ± 4), or (5, 3) and (5, –5). It has co-vertices at (5 ± 3, –1), or (8, –1) and (2, –1).

The major axis in a horizontal ellipse is given by the equation *y* = *v* ; the minor axis is given by *x* = *h*. The major axis in a vertical ellipse is represented by *x* = *h* ; the minor axis is represented by *y* = *v*. The length of the major axis is 2*a*. and the length of the minor axis is 2*b* .

You can calculate the distance from the center to the foci in an ellipse (either variety) by using the equation

where *F* is the distance from the center to each focus. The foci always appear on the major axis at the given distance (*F* ) from the center.

What if the elliptical equation you’re given isn’t in standard form? Take a look at the example

Follow these steps to put the equation in standard form:

Add the constant to the other side.

This gives you

Complete the square.

Determine if the ellipse is horizontal or vertical.

Because the bigger number is under *x*. this ellipse is horizontal.

Find the center and the length of the major and minor axes.

The center is located at (*h*. *v* ), or (–1, 2).

Graph the ellipse to determine the vertices and co-vertices.

Go to the center first and mark the point.

Source: m.dummies.com

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