### 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

### Why dothis problem?

In this problem, instead of giving the equations of some functions and asking learners to sketch the graphs, this challenge gives the graphs and asks them to find their equations. This encourages learners to experiment by changing the equations systematically to discover the effect on the graphs.

### Possible approach

Start by showing the picture and asking learners to identify the graphs of $y=x^2$ and $y=-(x-4)^2$.
Encourage discussion about the similarities and differences between these two graphs, and their equations.

Give learners time to experiment with graphical calculators or graphine software with the aim of idenitfying rules for modifying equations to transform graphs and create those on the image.

Draw ideas together with the purpose of defining a set of rules that can be applied in other similar situations.

A nice extension is for pairs to create a picture of their own, which they then challenge another pair to reproduce.

Those learners who are familiar with the idea of completing the square may wish to write the equation of each parabola in this form and consider the coordinates of the vertex.

You can read about one teacher's experience of using this task in the classroom.

### Key questions

• You are being asked to sketch a family of graphs. What makes this a family?
• What is the same and what is different about the equations $y=x^2$ and $y=-(x-4)^2$?
• How might these similarities and differences relate to the way they look and their positions on the axes?
• Can you convince us that the rules you have found will work with graphs of other functions.

### Possible extension

More Parabolic Patterns and Parabolas again offer similar pictures to reproduce. Cubics uses graphs of cubic functions, and Ellipses gives the opportunity to investigate the equation of an ellipse.

### Possible Support

Learners could begin by investigating translation of straight lines and look at how the equations change.

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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