Additional Useful Topics for the TMUA

These are additional or bonus topics that I think may be useful for TMUA preparation, particularly because some of the ideas are unfamiliar, overlooked, or not covered in much depth in the mainstream curriculum. They are not intended to be a comprehensive list of TMUA topics, and many of them are not especially high priority for the exam. For a much more comprehensive topic list, I highly recommend Tyler Tutoring. I have added a TMUA relevance score out of 10 to each topic to indicate how useful I believe it is specifically for TMUA preparation; this score is completely independent of difficulty. Use it to help decide which topics to study. Each topic is self-contained, so you can work through them in any order. If you spot an error or have a suggestion for improvement, please email me at jzmaths@hotmail.com.

1. 1Transformations: Part A — Core transformations and equations of curvesVertical and horizontal graph transformations, the order in which they combine, reflections in $y=b$ and $x=a$, and a first-principles method for transforming the equation of any curve.
2. 2Transformations: Part B — Modulus transformations, point mapping and rotationsThe modulus transformations $|f(x)|$ and $f(|x|)$, a direct shortcut for mapping a point to its image without drawing intermediate graphs, and rotating a curve through 90° about any point.
3. 3Multivariate Factorisation using the Factor TheoremLearn how to factorise multivariate polynomials using the factor theorem, including homogeneous polynomials and hidden factors, with techniques highly relevant to TMUA.
4. 4Cauchy-Schwarz and AM-GM InequalitiesLearn the Cauchy-Schwarz and AM-GM inequalities, with simple proofs, general forms, and examples showing when they can provide useful shortcuts in TMUA problem solving.