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TMUA 2023 · Paper 1 · Question 11 of 20

TMUA 2023 Paper 1 Question 11

Graphs and transformations — Transformations · matching minimum points. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Graphs and transformationsTransformations · matching minimum points6 options
It is given that f(x) = x2 − 6x.

The curves y = f(kx) and y = f(x − c) have the same minimum point, where k > 0 and c > 0.

Which of the following is a correct expression for k in terms of c?

  1. Ak = 3−c3
  2. Bk = 3c+3
  3. Ck = c−66
  4. Dk = 66−c
  5. Ek = c+99
  6. Fk = 9c−9
Show the answer and worked solution
answer · B
  1. Ak = 3−c3
  2. Bk = 3c+3
  3. Ck = c−66
  4. Dk = 66−c
  5. Ek = c+99
  6. Fk = 9c−9
f(kx) = k2x2 − 6kx has its minimum where 2k2x − 6k = 0, that is at x = 3k, and the value there is −9. f(x−c) = (x−c)2 − 6(x−c) has its minimum at x = c + 3, again with value −9. The y-coordinates already agree, so the points coincide when 3k = c + 3, giving k = 3c+3.