### Why do this problem?

This month's NRICH site
has been inspired by the way teachers at Kingsfield School in
Bristol work with their students. Following an introduction to a
potentially rich starting point, a considerable proportion of the
lesson time at Kingsfield is dedicated to working on questions,
ideas and conjectures generated by students.

This problem is based on
ideas that have been shared on the

NRICH Projects
discussion site. It links the idea of a geometric sequence with
analysis of a parabola and can lead to generalisations from the
graphs that can be proved algebraically.

### Possible approach

The activity could work
well with students working in small groups. Each group could take a
different geometric sequence and then formulate several equations
of the form $y=ax^2+2bx+c$, where $a$, $b$ and $c$ are three
consecutive terms from the sequence. After plotting the graphs, ask
students to comment on key similarities and differences between the
graphs.

Collect together each
group's findings on the board, noting down the sequence they chose
to use and any similarities and differences between the graphs they
noticed.

In order to prove any
conjectures the group suggests, some work on how to express a
general geometric sequence may be needed. There is ample
opportunity to practise factorising quadratic equations (including
those with coefficient of $x^2$ not equal to $1$) while working
towards an algebraic explanation for the patterns that occur.

### Key questions

What is the same about
each parabola?

What changes?

What happens when you try
different geometrical sequences?

### Possible extension

A very challenging
follow-up could be to ask students to explore cubic graphs where
terms from a geometric sequence could be substituted in to give
similar results.

### Possible support

Exploring Quadratic
Mappings provides introductory work on the equations of
parabolas.