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Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

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More Parabolic Patterns

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

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Parabolas Again

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?

Lattice Points

Stage: 5 Challenge Level: Challenge Level:1
The idea for this problem came from a Note in the Mathematical Gazette, Volume 20, July 2006 by Thomas Koshy entitled 'Lattice points in a family of hyperbolas'.

Even if you find some lattice points by trial and error you will still have to prove that there are no other possibilities.