Graph transformations — Scaled parabola on $(-\infty,3]$

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) Concave-down parabola; vertex when , : vertex . Domain takes the left half.
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) Set with . Swap: , . Then . Since , take the positive root: .
Finish the calculation, then check that the result meets the question’s conditions.Thus . Range of is , so domain of is . Equation: .
State the final answer clearly in the form the question requested.(iii) Reflect the graph from (i) in : endpoint , through (swap of ), through (swap of ), 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.