Graphs of Ellipses
Learning Objectives
- Sketch a graph of an ellipse centered at the origin
- Sketch a graph of an ellipse not centered at the origin
- Express the equation of an ellipse in standard form given the equation in general form
How To: Given the standard form of an equation for an ellipse centered at , sketch the graph.
- Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.
- If the equation is in the form , where , then
- the major axis is the x-axis
- the coordinates of the vertices are
- the coordinates of the co-vertices are
- the coordinates of the foci are
- If the equation is in the form , where , then
- the major axis is the y-axis
- the coordinates of the vertices are
- the coordinates of the co-vertices are
- the coordinates of the foci are
- If the equation is in the form , where , then
- Solve for using the equation .
- Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Example: Graphing an Ellipse Centered at the Origin
Graph the ellipse given by the equation, . Identify and label the center, vertices, co-vertices, and foci.Answer: First, we determine the position of the major axis. Because , the major axis is on the y-axis. Therefore, the equation is in the form , where and . It follows that:
- the center of the ellipse is
- the coordinates of the vertices are
- the coordinates of the co-vertices are
- the coordinates of the foci are , where Solving for , we have:
Therefore, the coordinates of the foci are . Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.
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