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## 'Geometric Parabola' printed from http://nrich.maths.org/

This problem developed from a question
posed on the NRICH
Projects site.
Oliver has been experimenting with quadratic equations of the form:
$$y=ax^2+2bx+c$$ Oliver chose values of $a, b$ and $c$ by taking
three consecutive terms from the sequence: $$1, 2, 4, 8, 16,
32...$$ Try plotting some graphs based on Oliver's quadratic
equations, for different sets of consecutive terms from his
sequence.

Do you notice anything interesting?

Can you make any generalisations? Can you prove them?

Oliver's sequence is an example of a geometrical sequence, created
by taking a number and then repeatedly multiplying by a common
ratio. Oliver's sequence starts at $1$ and has common ratio $2$
(each number in the sequence is $2$ times the previous number).

Create some more geometrical sequences and substitute consecutive
terms into Oliver's quadratic equation.

Here are some questions you might like to explore:

Can you make any predictions about the graph from the geometric
sequence you use to generate the equation?

What if the common ratio is a fraction, or a negative number?

What if the starting number for your geometric sequence is a
fraction, or a negative number?

Can you make any generalisations? Can you prove them?

You may wish to use graphing software
such as the free-to-download Geogebra to
investigate the graphs.