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Cubic Spin

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

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.

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.

More Parabolic Patterns

Age 14 to 18
Challenge Level

Sketch the graphof $y=x^2$. If you exchange $x$ and $y$ the point $(a,b)$ on one graph has an image $(b,a)$ on the other graph. What transformation of the graphs will have this effect? What do you expect the graph of $x= y^2$ to look like? Find these two graphs on the illustration.

Now use a graphic calculator, or graphing software, to sketch the two graphs. Were your predictions correct?

Find the other equations by considering the transformations of the graphs, changing the equations accordingly and testing your decisions by sketching the graphs and seeing if they match the graphs in the illustration in the way that you expected.