# Geometric Parabola

Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.

## Problem

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.

## Getting Started

$$y=ax^2+2bx+c$$

Start with $a=1, b=2, c=4$ - what does the graph look like? What key points does it pass through?

Then try $a=2, b=4, c=8$, $a=4, b=8, c=16$ and so on. Look for similarities and differences between the graphs.

Start with $a=1, b=2, c=4$ - what does the graph look like? What key points does it pass through?

Then try $a=2, b=4, c=8$, $a=4, b=8, c=16$ and so on. Look for similarities and differences between the graphs.

## Student Solutions

Patrick from Woodbridge school sent us this solution:

I plotted the first few equations with graphing software, and they all seemed to have the turning point at $(-2,0)$. I will now try and show this for all cases.

Taking $y=ax^2+2bx+c$, we can replace $a$, $b$ and $c$ by consective numbers from the geometic progression, say, $$\eqalign{a&=m\cr b&=2m\cr c&=2^2 m = 4m}$$ This gives us $$\eqalign{

y&=mx^2+2\times 2mx+4m\cr &= mx^2 + 4mx + 4m}$$

Now we differentiate to get $$\eqalign{\frac{dy}{dx}&= 2mx+4a \cr &= 2m(x+2)}$$

The turning point is given by $\frac{dy}{dx}= 0$, so $2m(x+2) = 0$ imples $x=-2$.

Substituting $x=-2$ into our equation for $y$ gives: $y=4m-8m+4m = 0$.

Therefore, all these curves have turning point at $(-2,0)$.

We can now generalise these ideas for any common ratio $n$.

As before we have: $$\eqalign{y&=ax^2 + 2bx + c\cr &= mx^2 + 2mnx + mn^2\cr &= m(x^2+2nx+n^2)}$$

Differentiating gives: $$dy/dx = 2mx + 2mn = 2m(x+n)$$

The x-coordinate of the turning point will thus be $-n$, and the y-coordinate is $(-n)^2 + 2n(-n) + n^2 = n^2 - 2n^2 + n^2 = 0$.

Therefore, for a quadratic in the form $y=ax^2+2bx+c$ where the coefficients are consequtive numbers in a geometric progression, the turning point of the curve will be $(-n,0)$. This will work regardless of whether n is natural or real.

## Teachers' Resources

### 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.

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

$$y=ax^2+2bx+c$$Start with $a=1, b=2, c=4$ - what does the graph look like? What key points does it pass through?

Then try $a=2, b=4, c=8$, $a=4, b=8, c=16$ and so on. Look for similarities and differences between the graphs.