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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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Cocked Hat

Sketch the graphs for this implicitly defined family of functions.

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Age 16 to 18 Challenge Level:

Why do this problem?
It requires an understanding of how to handle functions that are defined differently on different parts of their domains and how to interpret the situation when the derivative takes different values close to point but on opposite sides of the point.

Possible approach
A short problem that can be used as a lesson starter.

Key questions
What can you say about the function when $x< 0$?

What can you say about the function when $x> 0$?