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TMUA 2022 · Paper 1 · Question 12 of 20

TMUA 2022 Paper 1 Question 12

Algebra and functions — Family of quadratics · optimising over a parameter. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Algebra and functionsFamily of quadratics · optimising over a parameter5 options
A family of quadratic curves is given by yk = 2(x − k2)2 + k22 + 4k + 3 where k is any real number and yk is a function of x.

All these curves are sketched, and the point with the lowest y-coordinate among all the curves yk is (a,  b).

Find the value of a + b.

  1. A−1
  2. B−3
  3. C−5
  4. D−7
  5. E−9
Show the answer and worked solution
answer · D
  1. A−1
  2. B−3
  3. C−5
  4. D−7
  5. E−9
For a fixed k the curve is least at x = k2, where y = k22 + 4k + 3. Minimising that over k: its derivative is k + 4, zero at k = −4, giving y = 8 − 16 + 3 = −5 at x = −2. So (a, b) = (−2, −5) and a + b = −7.