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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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Cocked Hat

Sketch the graphs for this implicitly defined family of functions.

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Stage: 5 Challenge Level: Challenge Level:1

Drawing the graphs first will greatly aid in their interpretation!

Key to this problem is the idea for small values of $t$ we have $t$ is a lot larger than $t^2$, which is in turn a lot larger than $t^3$. For large values of $t$ the reverse is true.