In how many ways can you identify an equation for this parabola?

Below you will find four ways in which students have started to tackle this problem. For each approach, try to work out what they were doing and attempt to continue their work to reach a solution.

What is this student doing?

Why are they doing that?

What could they do next?

What is this student doing?

Why are they doing that?

What could they do next?

What is this student doing?

Why are they doing that?

What could they do next?

Student 4 has taken a more unusual approach. This approach is better considered after having explored one or more of the first three starting points.

\[\frac{dy}{dx}=2ax+b\]

What is this student doing?

Why are they doing that?

What could they do next?

You might like to think about how the approach taken by this student is connected to the ideas in Parabella.