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Powerful Quadratics

This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.


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

Factorisable Quadratics

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.

Can You Find... Trigonometry Edition

Age 16 to 18
Challenge Level

This resource is from Underground Mathematics.


Think of the types of graphs you can obtain by a combination of stretches, reflections and translations of the graph $y=\sin x$. In this resource we refer to any of these graphs as a "sine graph".

Can you find ...

(a) ... a sine graph which touches the lines $y=3$ and $y=1?$

(b) ... a cosine graph which crosses the $x$-axis at $x=1$ and $x=-1$?

(c) ... a tangent graph which passes through the point $\big(\dfrac{\pi}{3},0\big)$ and for which the line $x=\dfrac{\pi}{2}$ is an asymptote?

You can use the free Desmos graphing calculator to help you find suitable graphs, but try to sketch some graphs first.



Could you include extra conditions in parts (a), (b) or (c) so that the graphs are unique?




This is an Underground Mathematics resource.

Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.

Visit the site at to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.