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2023 HSC Mathematics Extension 1

Question 8:
Graph transformations

1 markgraph transformations

The diagram shows the graph of a function (nested absolute-value V-shapes with zeros at and local peaks of height at ).

Which of the following is the equation of the function?

The piecewise-linear graph supplied with the original question.
The piecewise-linear graph supplied with the original question.

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 graph is even about the -axis, so the equation must depend on (options C and D are shifted and fail symmetry about ).
  2. Zeros at and : for , set or . Matches.
  3. At : ; at : ; for large , grows linearly — matches the diagram.
  4. Option B: at , , but the graph is zero at .
  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

The graph with vertex opening upwards with slope has equation

See the step-by-step answer guide →

Question 2

For , the zeros occur at

  2. only
  4. no real zeros
See the step-by-step answer guide →

Question 3

Compared with , the graph of is obtained by

  1. reflecting the negative parts of in the -axis
  2. shifting unit right
  3. reflecting in the -axis only
  4. removing the absolute values
See the step-by-step answer guide →