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2021 HSC Mathematics Extension 1

Question 12(d):
Graph transformations

6 marksgraph transformationsdependency-safe group

A function is defined by for in the domain .

(i) Sketch the graph of showing the - and -intercepts.

(ii) Find the equation of the inverse function , and state its domain.

(iii) Sketch the graph of .

How to recognise this question

Question type: graph transformations

  • The command and notation point to graph transformations.
  • The word “hence” means the later part is intended to reuse the earlier result.

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

Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.

Step-by-step answer

  1. (i) is a concave-down parabola (negative coefficient). Vertex when , : vertex .
  2. Domain takes only the left half of the parabola.
  3. -intercept: , so .
  4. -intercepts: or or . Only lies in the domain, so intercept .
  5. NESA (i): 2 marks; 1 for left half of a concave-down parabola, or a full parabola with one intercept/vertex.
  6. (ii) Set with . Swap: , .
  7. . Because , take the positive root: .
  8. Thus .
  9. Range of is , so domain of is . Equation: .
  10. NESA (ii): 3 marks; partial for expression alone, or domain with incomplete inversion.
  11. (iii) Reflect the graph from (i) in : endpoint , through (swap of ), through (swap of ), domain .
  12. NESA (iii): 1 mark for a correct inverse sketch.

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

The paper records 6 total marks 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

A function is defined by for in the domain .

(i) Sketch the graph of showing the - and -intercepts.

(ii) Find the equation of the inverse function , and state its domain.

(iii) Sketch the graph of .

See the step-by-step answer guide →

Question 2

A function is defined by for in the domain .

(i) Sketch the graph of showing the - and -intercepts.

(ii) Find the equation of the inverse function , and state its domain.

(iii) Sketch the graph of .

See the step-by-step answer guide →

Question 3

A function is defined by for in the domain .

(i) Sketch the graph of showing the - and -intercepts.

(ii) Find the equation of the inverse function , and state its domain. Hence,

(iii) Sketch the graph of .

See the step-by-step answer guide →