There are many different approaches to this problem. Here are the questions Emma, Tamsin, Ellie, Sophie, Isabella and Alex from Sandbach High school used to help narrow down the possible choice of quadratics.
Is it a negative quadratic shape?
A
Is the turning point on the y-axis?
Is the turning point on the x-axis?
B
Has it been translated vertically?
Has it been translated horizontally?
Though group A and B are similar questions.
They have also observed that group A and group B are very similar questions, so in fact you only need to ask one of these pairs of questions. Can you see why? How would the answers to one of the pairs of questions give you the answers to the other pair and vice versa?
From this point we still need to ask a few more questions to identify the quadratic exactly out of the 18 possibilities. Nicolette from St Stephen's School has suggested some more specific questions to end with:
More specific questions could be if the parabola has been vertically translated upwards or downwards, or if the parabola has been translated horizontally left or right.