Graph transformations — Reciprocal of a simple cubic shape

Question

The graph of has zeros at and , is positive on , and has a maximum value at . Describe the key features of .

This practice question was inspired by Question 11(c) in the 2020 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.Vertical asymptotes at and .
Simplify the previous line carefully, keeping exact values where possible.On , .
Finish the calculation, then check that the result meets the question’s conditions.At the max of , has a local minimum .
State the final answer clearly in the form the question requested.As at infinity (if so), .

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

Verification: Asymptotes at the zeros; local max of becomes local min of when .

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