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Ellipses
Age 14 to 18
Challenge Level 
Consider the symmetries of the graphs in this family, where the
graphs cut the axes, and how they can be thought of as stretched
circles. Relate these ideas to the equations you get by taking
different values of the constants $a$ and $b$ in the equation
$${x^2\over a^2} + {y^2\over b^2}=1.$$ All these ideas come
together to throw light on transformations of graphs and also on
properties of ellipses.
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