Graph transformations — Scaled parabola on $(-\infty,4]$

Question

A function is defined by for in the domain .

(i) Sketch the graph of showing the - and -intercepts.

(ii) Find the equation of the inverse function , and state its domain.

(iii) Sketch the graph of .

This practice question was inspired by Question 12(d) in the 2021 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: track how each transformation moves key points, asymptotes and intercepts of the base graph.(i) Vertex when , : vertex . Left half of a concave-down parabola.
Simplify the previous line carefully, keeping exact values where possible.-intercept: , so . -intercepts: or ; only in the domain, so .
Simplify the previous line carefully, keeping exact values where possible.(ii) , . Swap and take (positive because ). Hence .
Finish the calculation, then check that the result meets the question’s conditions.Domain of is the range of , namely . So on .
State the final answer clearly in the form the question requested.(iii) Reflect (i) in : endpoint , through and , domain .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: (i) Vertex , intercepts , . (ii) on . (iii) Reflection of (i).

Any mark labels are a TestMum study aid, not an official NESA marking allocation.