TestMum
← 2022 HSC Extension 1 paper Question 4

Variation 3 · answer guide

Graph transformations — Turning points from components

Question

Suppose is strictly increasing and is strictly decreasing. Which can their sum still have?

  1. A local maximum (if the decrease of dominates then loses)
  2. No turning points ever
  3. Only discontinuities
  4. Infinitely many vertical asymptotes forced by the sum alone

This practice question was inspired by Question 4 in the 2022 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Step by step

  1. Set up the solution: track how each transformation moves key points, asymptotes and intercepts of the base graph.Derivatives add: can change sign even if each factor does not vanish.
  2. Finish the calculation, then check that the result meets the question’s conditions.So a local max/min of the sum is possible.
  3. Match the result to the choices and state the correct option.Therefore option A is correct.

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

Verification: The sum's derivative is not forced to keep a fixed sign.

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

← Back to 2022 HSC Extension 1 paper Question 4