Stage: 5 Challenge Level: 
The implicit equation
$$(x^2 + 2a y -a^2)^2 = y^2(a^2 - x^2)$$
represents a family of functions the members of the family corresponding to different values of $a$.
Re-arrange the equation into the form $y= f(x)$ and draw some graphs belonging to the family for different values of $a$. What is the domain of the function $f(x)$?
For investigating these graphs you can download the shareware
program Graphmatica for free from here as
NRICH is an approved distributor of this program. You can find
more information about the program from http://www.graphmatica.com/ .
Families of Graphs. Mathematical reasoning & proof. Surds. Long problems. Manipulating algebraic expressions/formulae. Symmetry. Complex numbers. Graph sketching. Golden ratio. Quadratic equations.
Published January 2003.
Copyright © 2007. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project, which also includes the Plus and Motivate sites.
email: nrich@damtp.cam.ac.uk
