### 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.

### 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.

# Parabolic Patterns

##### Stage: 4 and 5 Challenge Level:

The illustration shows the graphs of fifteen functions. Two of them have equations

$y = x^2$
$y = - (x - 4)^2$

Use a graphic calculator or a graph drawing computer program to sketch these two graphs and then locate them in this illustration. Use the clues given in this information to help you to find the equations of all the other graphs and to draw the pattern of the 15 graphs for yourself. For your solution send in the equations you have found with an explanation of how you did it.

What about the equations of these parabolas?

You may like to use your creative talents to devise your own pattern of graphs and send them to us so that we can base another challenge like this one on the website using your pattern.

NOTES AND BACKGROUND
This sort of challenge is sometimes called an inverse problem because the question is posed the opposite way round to what might have been expected. This is almost like saying: 'here is the answer, what was the question?' Instead of giving the equations of some functions and asking you to sketch the graphs, this challenge gives the graphs and asks you to find their equations.

You are being asked to sketch a family of graphs. What makes this a family? All the graphs are obtained by transformations such as reflections and translations of other graphs in the family. The key is to find the simplest function and then tofind transformations of the graph of that function which give the other graphs in the family.

If you have access to a graphic calculator, or tograph drawing software, it will not give you the answers. You will have to think for yourself what the equations should be and then the software will enable you to test your own theories and see if you were right.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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