TestMum
← 2022 HSC Extension 1 questions

2022 HSC Mathematics Extension 1

Question 5:
Coordinate geometry

1 markcoordinate geometry

A curve is defined in parametric form by and for .

Which diagram best represents this curve?

All four directed curve options supplied with the original question.
All four directed curve options supplied with the original question.
  1. Arc from to with open endpoints and concave-up appearance
  2. Arc from to with filled endpoints, concave down (left half of a downward parabola)
  3. Arc from to with an open lower-left mark and concave-up shape
  4. Arc that passes and continues past to the right

How to recognise this question

Question type: coordinate geometry

  • The command and notation point to coordinate geometry.
  • The requested response is a multiple choice.

How to handle it: convert between parametric and Cartesian forms carefully, noting the direction of the parameter.

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

Step-by-step answer

  1. Endpoints: gives ; gives . The closed interval includes both endpoints (filled).
  2. Eliminate : , so , a downward-opening parabola with vertex .
  3. On we have : only the left-hand half of the parabola, ending at the vertex.
  4. As increases from to , the point moves from to (rightward and upward).
  5. Option B matches filled endpoints, concave-down arc, and terminates at without continuing past the vertex. Answer: B.

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

For , with , which description is correct?

  1. From to , filled endpoints, concave down
  2. From to , concave down
  3. From to , open endpoints
  4. Full parabola through both sides of
See the step-by-step answer guide →

Question 2

On , for , the particle moves

  1. from to , left half of an upward parabola, endpoints included
  2. from to
  3. along the full parabola
  4. vertically only
See the step-by-step answer guide →

Question 3

If instead for , , the diagram should show

  1. open endpoints near and on the same arc
  2. filled endpoints at and
  3. the entire infinite parabola
  4. only the single point
See the step-by-step answer guide →