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.

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.

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.

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

Quadratic graphs are very familiar, but what patterns can you explore with cubics?

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

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

Use functions to create minimalist versions of works of art.

Observe symmetries and engage the power of substitution to solve complicated equations.

Draw graphs of the sine and modulus functions and explain the humps.

Step back and reflect! This article reviews techniques such as substitution and change of coordinates which enable us to exploit underlying structures to crack problems.

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?