Transformation of functions

article
The why and how of substitution

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.
problem
Sine problem

Age

16 to 18

Challenge level

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.
problem
More parabolic patterns

Age

14 to 18

Challenge level

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.
problem
Ellipses

Age

14 to 18

Challenge level

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

Age

14 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

14 to 18

Challenge level

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.
problem
Exploring cubic functions

Age

14 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

Use functions to create minimalist versions of works of art.