Functional notation and transformations is always tricky to teach and understand. GCSE students will meet this in Year 11. In general, transformations applied after the function are more easily understood:
Something very un-intuitive happens when the transformation is applied to the argument of the function:
with things stretching when they look like they should be compressing and other things moving the wrong way.
Mathematics is not always obvious, if it were we wouldn’t need it.
Try this for transformations of Sine.
The sine function starts life in a triangle (year 9?) then becomes intimate with a circle at A level, as y-coord of a point rotating on a unit circle. Later it is about infinite series or, even, differential equations.
Carr’s 626, in his magical book for Edwardian mathematical autodidacts, provides summary of general solutions for trigonometrical equations useful to STEP and A level students.
Friendly triangles – good visual method for memorising trig ratios of 30, 45 and 60 degree angles. Known for thousands of years as per the Babylonian tablet