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?

How many different ways do you know to write the equation of a quadratic?

Which of these might be most useful for each of these problems?

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 \(A\)?

Which form of quadratic most easily allows you to find the vertex of the quadratic?

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.

It will be useful when working on this one to use clear notation for the coordinates of \(A\), \(B\) and \(C\). Something like \(A(a_1,a_2)\), \(B(b_1,b_2)\) and \(C(c_1,c_2)\) would work well.

If the quadratic is \(y=px^2+qx+r\) (using \(p\), \(q\) and \(r\) instead of the usual \(a\), \(b\) and \(c\) to avoid confusion), what equations can we write down for \(p\), \(q\) and \(r\)? How can we go about solving these?

Can you find a quadratic which passes through \(A(a_1,a_2)\) and intercepts the \(x\)-axis at \((b_1,0)\) and \((c_1,0)\)?

What other two similar quadratics can you find?

How can you combine these to solve the problem?