EMAT 6680

Assignment 2

Stephanie K. Lewis

10. Graph x² + y² = 9

On, the same axes, graph x² + xy + y² = 9

Describe the new graph.

The new graph is an ellipse with x-intercepts at (-3,0) and (3,0) and y-intercepts at (0,3) and (0, -3). The lines of symmetry appear to be the lines y = x and y = -x

Try different coefficients for the xy term. What kinds of graphs do you generate?

If we let a represent the x term coefficient, b represent the y term coefficient, and c represent the constant, then the following observations can be made. In general,

1. If a and b are equal, then the graph will be a circle.

   Let a = 2 and b = 2. Then the graph of 2x² + 2y² = 9 (in purple) and the graph of 2x² + (2)(2) + 2y² = 9 (in red) are illustrated below.

   

    

2. If a, b, and c have the same signs, and a and b are unequal, then the graph will be an ellipse.

   Let a = 2 and b = 4. Then the graphs of 2x² + 4y² = 9 (in purple), and 2x² + (2)( 4) + 4y² = 9 (in red) are illustrated below.

   

    

   3. If a and b have opposite signs, then the graph will be a hyperbola.

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