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2022 HSC Mathematics Extension 1

Question 2:
Graph transformations

1 markgraph transformations

The graph of is shown. (Accessible description: vertical asymptote , horizontal asymptote ; for the graph is above and decreases toward it; for it lies below and approaches the asymptotes.)

The graph of was transformed to get the graph of as shown. (Accessible description: is even about the -axis, with vertical asymptotes , and for it matches the right-hand branch of reflected to the left.)

What transformation was applied?

The supplied graph of f and the transformed graph of g.
The supplied graph of f and the transformed graph of g.

How to recognise this question

Question type: graph transformations

  • The command and notation point to graph transformations.
  • The requested response is a multiple choice.

How to handle it: track how each transformation moves key points, asymptotes and intercepts of the base graph.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. The transformed graph is symmetric about the -axis: the right-hand branch of is copied to the left of .
  2. Replacing by leaves the graph unchanged for and reflects the graph across the -axis for . That is exactly .
  3. Option C only reflects through the -axis (swaps left and right, does not create even symmetry from an asymmetric ).
  4. Option B (square root) and option D (reciprocal) change heights and asymptotes, not merely mirror the positive- branch.
  5. Answer: A. (NESA multiple-choice key: A.)

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.

Try your hand at it again by answering these questions

Question 1

Starting from a non-even graph defined for with a vertical asymptote at , which transformation produces an even graph with asymptotes at ?

See the step-by-step answer guide →

Question 2

If has a single vertical asymptote at (and is not even), which of the following produces a graph with a single asymptote at ?

See the step-by-step answer guide →

Question 3

For , which statement about is true?

  1. is even with asymptotes
  2. has a zero where has a vertical asymptote
  3. has a horizontal asymptote and is not the even graph shown in the original question
  4. is identical to
See the step-by-step answer guide →