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,
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.
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.