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Quartics

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

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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.

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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

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Expand the square the left hand side of the expression and rearrange the expression to give a quadratic equation in $y$. Now solve the quadratic equation. This will give two solutions (as quadratic equations generally do) and each will give you the equation of a branch of the graph. Sketch the graph by considering where real values of $y$ exist, where the graph crosses the axes and what symmetries it has. You can use graphing software but it is much more of a challenge to sketch the graph without.

Don't let this defeat you. It is not nearly as hard as it looks but you have to do a little algebra. If you wish you can choose a particular value for $a$, say $a=1$, and investigate one member of the family of graphs but then you will have to go back and see what happens for different values of $a$. You can use graphing software but it is much more of a challenge to sketch the graph without.