### 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.

# Cubics

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

Many of you have sent in absolutely stunning graph patterns that you have created using a single graph and transformations of it. You have discovered how to translate the graph and how to stretch it in different directions by changing the equation of the graph accordingly.

This is Ali Abu-Hijleh's pattern. We gave you two of the equations: $$y = (x+6)^3 - 2$$ $$y=-(x-9)^3+3$$and here are the equations of the other curves: $$y=x^3$$ $$y=-x^3$$ $$y=-(x+9)^3-3$$ $$y=(x+9)^3-3$$ $$y=-(x+6)^3-2$$ $$y=(x+6)^3-2$$ $$y=-(x+3)^3-1$$ $$y=(x+3)^3-1$$ $$y=-(x-3)^3+1$$ $$y=(x-3)^3+1$$ $$y=-(x-6)^3+2$$ $$y=(x-6)^3+2$$ $$y=-(x-9)^3+3$$ $$y=(x-9)^3+3$$Well done Hsiu Chen; Claire from the Mount School, York; Malcolm and Charles from Madras College, St Andrew's and Andrei from School No. 205, Bucharest, Romania. All these students gave the equations with very good explanations of their methods as follows:

I started from the graph of the function: $$y=x^3$$ then I identified it on the pattern. I saw that the reflection of this graph, either in the $x$-axis or in the $y$ axis, gives: $$y=-x^3%$$I displaced these graphs along the $x$ and $y$ axes. I identified the scale by looking at the two equations given in the problem. The scale in the drawing is: $x$ between $-12$ and $12$, and $y$ between $-6$ and $6$.

All the equations are of the form $-y = \pm (x\pm a)^3\pm b$ where adding and subtracting $b$ translates the graph up and down parallel to the y-axis and changing $x$ to $x\pm a$ translates the graph parallel to the x-axis. It happens that in this example we always have $a=3b$.