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2024 HSC Mathematics Extension 1

Question 5:
Inverse trigonometric functions

1 markinverse trigonometric functions

Consider the function .

Which transformations have been applied to to obtain ?

  1. Vertical dilation by a factor of and a horizontal dilation by a factor of
  2. Vertical dilation by a factor of and a horizontal dilation by a factor of
  3. Vertical dilation by a factor of and a horizontal dilation by a factor of
  4. Vertical dilation by a factor of and a horizontal dilation by a factor of

How to recognise this question

Question type: inverse trigonometric functions

  • The command and notation point to inverse trigonometric functions.
  • The requested response is a multiple choice.

How to handle it: use the restricted domain and range that make the inverse function single-valued.

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

Step-by-step answer

  1. From to : replace by , which is a horizontal dilation by factor (compression toward the -axis).
  2. Then multiply by : is a vertical dilation by factor .
  3. Option A reverses both factors; B and D get the horizontal factor wrong (using instead of ).
  4. Answer: C. (NESA multiple-choice key: C.)

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

Compared with , the graph of is obtained by a horizontal dilation of factor

See the step-by-step answer guide →

Question 2

Transformations taking to are

  1. vertical factor , horizontal factor
  2. vertical factor , horizontal factor
  3. vertical factor , horizontal factor
  4. vertical factor , horizontal factor
See the step-by-step answer guide →

Question 3

From to , the vertical dilation factor is

See the step-by-step answer guide →