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2022 HSC Mathematics Extension 1

Question 4:
Graph transformations

1 markgraph transformations

The diagram shows the graph of the sum of the functions and . (Accessible description of : a continuous curve that is negative for large negative , crosses the negative -axis, has a local maximum above the -axis left of the origin, dips through a local minimum on the positive -axis side, then rises again.)

Which of the following best represents the graphs of both and ?

The supplied sum graph and all four candidate pairs of component graphs.
The supplied sum graph and all four candidate pairs of component graphs.
  1. Option A: a pair of component graphs whose pointwise sum matches the given sum (one roughly linear-increasing through the origin region and one with a single hump/trough consistent with the sum's turning points)
  2. Option B: a pair whose sum would have the wrong number or location of turning points
  3. Option C: a pair whose sum would have incorrect sign pattern for large
  4. Option D: a pair whose sum would not reproduce the given intercept/turning structure

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 sum graph's key features (signs far left/right, number and order of turning points, axis intercepts) must equal at corresponding .
  2. Check candidate pairs at several anchor -values: where both are positive the sum is above either; where they have opposite signs the sum is between them.
  3. Only option A produces the same sequence of local max/min and the same end behaviour as the given sum diagram.
  4. Options B–D fail at least one of: turning-point count, intercept location, or asymptotic sign.
  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 of is a parabola opening upward with vertex on the -axis. Which pair is consistent?

  1. , (constant)
  2. , (sum zero)
  3. ,
  4. ,
See the step-by-step answer guide →

Question 2

If and , the sum graph is

  1. the horizontal line
  2. the line
  3. the line
  4. the parabola
See the step-by-step answer guide →

Question 3

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
See the step-by-step answer guide →