How to Find the Vertices & Graphing Ellipses

Ellipses are geometric shapes found in math, nature and construction.

An ellipse has two axes which are perpendicular to each other and intersect at the ellipse's center. The major axis is the longest axis. Its endpoint's are the ellipse's vertices. The minor axis contains the ellipse's co-vertices. The coordinates of the vertices and co-vertices are important in graphing an ellipse because they lie on the surface of the ellipses. If you are given the equation of an ellipse, you can graph the ellipse by first finding the vertices and co-vertices using algebraic techniques.

Instructions

1. Determine the major axis of your ellipse. The standard equation for an ellipse is "x^2/a^2 + y^2/b^2 = 1" In this equation, if "a" is greater than "b," the major axis of your ellipse is horizontal or the x axis. If "a" is less than "b," however, the major axis of your ellipse is vertical or the y axis.

2. Calculate the vertices on the major axis. If you determined your major axis to be horizontal, solve for "x" in the equation of the ellipse while setting "y" equal to 0. If you determined your major axis to be vertical, solve for "y" while setting "x" equal to 0. Use the equation "x^2/16 + y^2/9 = 1" as an example. In this equation "16" is equal to "a^2" and "9" is equal to "b^2." The major axis is horizontal because "a" is greater than "b." After setting "y" equal to 0 the equation becomes "x^2/16 = 1." Multiply both sides of the equation by 16 and take the square root of both sides. You will find that "x" is equal to 4 or -4. These are the two vertices (4,0) and (-4,0).

3. Find the co-vertices. If you determined your major axis to be horizontal, solve for "y" in the equation of the ellipse while setting "x" equal to 0. If you determined your major axis to be vertical, solve for "x" while setting "y" equal to 0. After solving for "y" in the example equation, you will find that the vertices on the minor axis are (0,3) and (0,-3).

4. Plot the vertices on a graph using the coordinates found in steps 2 and 3.

5. Connect the vertices using smooth curved lines.

References

• Math Warehouse: Equation of an Ellipse in standard form and how it relates to the graph of an ellipse
• Wolfram MathWorld: Ellipse