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TMUA 2022 · Paper 1 · Question 10 of 20

TMUA 2022 Paper 1 Question 10

Graphs and transformations — Translations of a cubic. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Graphs and transformationsTranslations of a cubic8 options
A sequence of translations is applied to the graph of y = x3.

Which of the following graphs could be the result of this sequence of translations?

I   y = x3 − 3x2 + 9x − 27
II  y = x3 − 9x2 + 27x − 3
III y = 27x3 − 9x2 + x − 3

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · C
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
A translation of y = x3 has the form y = (x−h)3 + k = x3 − 3hx2 + 3h2x − h3 + k. The leading coefficient must stay 1, which rules out III. In I the x2 term gives h = 1, so the x term should be 3, not 9 — so I fails. In II the x2 term gives h = 3 and the x term is indeed 3(9) = 27, with k = 24 fixing the constant. Only II.