Coordinate Geometry and Graphs

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.

This is a beautiful result involving a parabola and parallels.

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

Can you work out the equations of the trig graphs I used to make my pattern?

Which curve is which, and how would you plan a route to pass between them?

Can you find a quadratic equation which passes close to these points?

Which line graph, equations and physical processes go together?