The Parabola

Everything Quadratic

The parabola – probably the best curve in the world.

First analysed by the ancient Greeks in the middle fourth century BC, and put to use in earnest by Galileo who was interested in the trajectories of cannonballs. Parabolic shapes are found all over nature, architecture, engineering, and they are all described by a quadratic equation.

During this activity we are going to look at graphs of parabolas, and derive the actual equations of curves from a few single points on the graph. You’re going to think about quadratic equations, hopefully in all their forms, maybe also transformations, completing the square, quadratic sequences…. So much beautiful maths. Let’s get started.

WARNING! These tasks start quite easy but there are some very challenging tasks here, (some of our A level students might struggle). Have a go and if you can't see the solution, check the solutions provided and see if you can work out where it comes from. There are some fun activities at the end!

Quadratic Equations: A quick reminder

You should have come across three different ways of writing quadratics. Each of them is useful and interesting in their own way.

We’re going to be using all three forms today, and hopefully getting a feel for how the equations relate to the curves. We have looked at quadratic equations and graphs back in week 2 if you would like to refresh your memory.

Week 2

Give Me Three Points and I’ll Give you a Curve

Can you find the equation of this blue curve that passes through the three points shown?

Solution is here

3 More sets of points

Hint/Solution

You could think about transformations, or vertex form. Even better, think about both!

Solution

3 more sets... same hint as the previous puzzle!

Solution

Now combine the last 2 problems to solve this one

Can you generalise to move this to any 3 points in this pattern?

Solution

Root Fitting

If we know the x intercepts, we know a lot about our equation, but not everything.

So we know the roots and the y-intercepts, what are the equations?

Hint/Solution

The factorised form tells us about the roots (x-intercepts). Can you combine that with transformations?

Solution

And now generalise: Can you find a solution wherever your points lie on the axes?

Here you can move all three points around, and your quadratic should still match.

Solution

Challenge!

It gets a bit difficult if our third point is somewhere else on the graph, not the y intercept? Can you find a solution here?

You might like to try your own example: Choose your three points with some nice numbers to play with, and get that one to work before you try this general solution.

3 in a Row

What strategy might you use if our three points look like this?

What can we do with these three points? Can you find the equation of the parabola?

Hint/Solution

Can you think of the y values as the first 3 terms in a Quadratic Sequence?

Solution

Challenge! Can you find the quadratic for 3 sequential points like this wherever they are? i.e. the points are (a,1) (a+1, 2) (a+2, 5)

Solution

These three points are all at regular intervals too:

Can you change the previous equation we found to find this equation?

Solution

Super Challenge

ANY 3 Points

If we know any 3 points, there is ONLY ONE quadratic equation that will fit all three. However finding this equation isn’t easy. Have a go at this example if you want a challenge.

Hint/Solution

3 simultaneous equations?

Solution

Art Time - Make the curve match the picture

Click the picture to open the graph in Desmos and match all the parabolas you can find.

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