### 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

### Why do this problem?

Many maths courses focus in great detail on quadratic functions and their graphs, but cubic functions are often only touched on briefly. This problem takes some particular examples of cubic functions and invites students to explore their graphs.

By switching between different algebraic representations and the graphical representation, students can gain insights that would not be so easily apparent if working purely algebraically. This, along with the scaffolding provided by the interactive GeoGebra applets, allows students to become more resilient in their approach to curve sketching, a topic which many find challenging.

### Possible approach

Students will get the most benefit from working on this task if each group has access to tablets or laptops so they can explore the GeoGebra applets. Alternatively, if only one computer and a projector is available, this could be used as a whole class activity, with the teacher moving the sliders and inviting students to notice what happens and make predictions.

Invite students to move the slider on the first applet to change the function, and to write down what they notice. Use the questions in the problem as prompts:
What stays the same?
What changes?
What might the function be?

Draw attention to the roots of the cubic, and the relationship between the function $f(x)=x(x-a)(x+a)$ and the shape of the graph. Invite students to expand the function.

Now move on to the second applet and go through the same process - move the sliders, notice what stays the same and what changes, and then make connections between the key points of interest on the curve (roots, turning points, y-intercept) and the function.

Once students have had plenty of time to explore the applets and gain insights into the relationship between the functions and their graphs, hand out this worksheet and challenge students to match the functions with the graphs. You could finish off the lesson by inviting each group to explain their reasoning.

The final activity in the problem is available on this worksheet, which could be used as a consolidation activity in another lesson or for homework.

### Possible support

Students could start by exploring quadratic graphs before moving on to cubics. Parabolic Patterns provides a nice challenge that uses similar thinking.

### Key questions

Where does the graph cross the x axis?
Where does it cross the y axis?
Does the graph have any reflectional or rotational symmetry?
Does the graph have any turning points?

### Possible extension

Students could explore the graphs of Ellipses in a similar way.