You're invited to decide whether statements about the number of solutions of a quadratic equation is always, sometimes or never true.

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.

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?

Can you sketch and then find an equation for the locus of a point based on its distance from two fixed points?

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?

Can you find values that make these surd statements true?

If you know some information about a parabola, can you work out its equation?

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