### Why do this problem?

This

problem begins to train students in the issue of fitting curves to data, so important in the applications of mathematics. It is a good problem to do when considering turning points of polynomials.

### Possible approach

This problem would initially suit a discussion approach. Why would certain curves definitely not work? This would draw students into an appreciation of the importance of turning points in this problem.

The next phase will be to draw students to the idea that a general cubic equation takes a certain form and that this form will be restricted by the constraints.

### Key questions

What sorts of functions would obviously have no chance of working? Why?

What sorts of functions might have a chance of working? Can we say why?

How could we impose the constraints algebraically?

### Possible extension

Here we specify one point and two turning points and this 'over-constrains' the problem so that we cannot fit a cubic. As an extension, students could think about how many points and turning points we can specify and be sure to be able to fit a polynomial of degree $n$.

### Possible support

Look at the hint.