### 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.

# Exploring Cubic Functions

##### Age 14 to 18 Challenge Level:

Adithya from Hymers College in the UK sent in some observations about the first GeoGebra applet:
Similarities between all the 11 cubic graphs as you move the slider include the fact that they all pass through the origin $( 0,0 )$ and therefore all have a y-intercept of $0,$ they are all positive cubics and the sum of their three roots is $0.$ In addition, where $1 \leq a \leq 10$ on the slider, the cubic graphs have turning points known as local maxima and local minima.

Furthermore, all these cubic graphs are rotationally symmetric given that the maximum point is rotated to become a minimum and vice versa.

Differences between the cubic graphs include their different roots, and the graph of $y = x^3,$ occurring when $a = 0$ on the slider, has stationary point known as a point of inflection at $x = 0$ rather than local maxima and local minima areas which are present in the other ten cubic graphs.

Jacqueline and Stella from Sandbach High School in the UK and Adithya matched the curves A - F to the equations 1 - 6. This is Jacqueline and Stella's working:

Adithya found the equations of the family of cubics:

Adithya also sent in some observations about the second applet, which link the algebraic and graphical features of cubics. Click below or here to see Adithya's work, with some notes added.