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:

or

Something very un-intuitive happens when the transformation is applied to the argument of the function:

or

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.

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