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TMUA 2021 · Paper 1 · Question 18 of 20

TMUA 2021 Paper 1 Question 18

Graphs and transformations — Transformations · rotating a curve about a point. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1Graphs and transformationsTransformations · rotating a curve about a point8 optionshard
The curve with equation x = y2 − 6y + 11 is rotated 90∘ clockwise about the point P to give the curve C.

P has x-coordinate −2 and y-coordinate 3.

What is the equation of C?

  1. Ay = −x2 − 4x − 3
  2. By = −x2 − 4x − 5
  3. Cy = −x2 − 6x − 7
  4. Dy = −x2 − 6x − 11
  5. Ey = x2 − 4x + 5
  6. Fy = x2 + 4x + 3
  7. Gy = x2 − 6x + 11
  8. Hy = x2 + 6x + 7
Show the answer and worked solution
answer · B
  1. Ay = −x2 − 4x − 3
  2. By = −x2 − 4x − 5
  3. Cy = −x2 − 6x − 7
  4. Dy = −x2 − 6x − 11
  5. Ey = x2 − 4x + 5
  6. Fy = x2 + 4x + 3
  7. Gy = x2 − 6x + 11
  8. Hy = x2 + 6x + 7
A clockwise quarter-turn about P(−2, 3) sends a point (x,  y) to (−2 + (y−3),  3 − (x+2)), since the displacement (a,  b) from P becomes (b,  −a). Writing the image as (X,  Y) gives X = y − 5 and Y = 1 − x, so y = X + 5 and x = 1 − Y. Substitute into the original, written as x = (y−3)2 + 2: 1 − Y = (X+2)2 + 2. Rearranging, Y = −(X+2)2 − 1 = −X2 − 4X − 5.