GeoGebra constructions... quadratic edition

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Two points on the axis

Using GeoGebra, plot two points \(A\) and \(B\) on the \(x\)-axis of a graph. Can you instruct GeoGebra to draw a quadratic which passes through both of them?

It should be constructed in such a way that when you move either \(A\) or \(B\) along the \(x\)-axis, the quadratic changes so that it continues to pass through both of them.

If you want some guidance on how to get the coordinates of the points \(A\) and \(B\), or how to enter an equation in GeoGebra, see the Using GeoGebra section.

This applet shows what the result could look like:

Taking it further

Developing this idea further, add a point \(C\) to the \(y\)-axis. Can you make your quadratic pass through \(A\), \(B\) and \(C\), like this?

One point as the vertex

On a new graph, plot a point \(A\). Can you instruct GeoGebra to draw a quadratic which has its vertex at \(A\)?

It should be constructed in such a way that when you move \(A\), the quadratic changes in response.

This is what the result could look like:

Taking it further

Developing this idea further, add a point \(B\) to the \(y\)-axis. Can you make your quadratic have its vertex at \(A\) and pass through \(B\), like this?

Any three points

The final (very hard) challenge: plot three points \(A\), \(B\) and \(C\) anywhere on the graph, and draw a quadratic passing through these three points, like this:

Is there more than one quadratic you could draw?