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Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Sine Problem

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

Folium of Descartes

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Cocked Hat

Age 16 to 18 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 $f(x)$?

To investigate these graphs you might try