In this activity you will need to work in a group to connect different representations of quadratics.

How does the temperature of a cup of tea behave over time? What is the radius of a spherical balloon as it is inflated? What is the distance fallen by a parachutist after jumping out of a plane? After sketching graphs for these and other real-world processes, you are offered a selection of equations to match to these graphs and processes.

This gives you an opportunity to explore roots and asymptotes of functions, both by identifying properties that functions have in common and also by trying to find functions that have particular properties. You may like to use the list of functions in the Hint, which includes enough functions to complete the table plus some extras.You might like to work on this problem in a pair or small group, or to compare your table to someone else's to see where you have used the same functions and where not.

Here you have an expression containing logs and factorials! What can you do with it?

A lune is the area left when part of a circle is cut off by another circle. Can you work out the area?

What graphs can you make by transforming sine, cosine and tangent graphs?

Here are some more triangle equations. Which are always true?

Can you make sense of these unusual fraction sequences?

What questions does the spiral construction prompt you to ask?

Can you find trig graphs to satisfy a variety of conditions?