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TMUA 2019 · Paper 1 · Question 1 of 20

TMUA 2019 Paper 1 Question 1

Algebra and functions — Quadratics · completed square from a turning point. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1Algebra and functionsQuadratics · completed square from a turning point6 options
f(x) is a quadratic function in x.

The graph of y = f(x) passes through the point (1,  −1) and has a turning point at (−1,  3).

Find an expression for f(x).

  1. A−x2 − 2x + 2
  2. B−x2 + 2x + 3
  3. Cx2 − 2x
  4. Dx2 + 2x − 4
  5. E2x2 + 4x + 1
  6. F−2x2 − 4x + 5
Show the answer and worked solution
answer · A
  1. A−x2 − 2x + 2
  2. B−x2 + 2x + 3
  3. Cx2 − 2x
  4. Dx2 + 2x − 4
  5. E2x2 + 4x + 1
  6. F−2x2 − 4x + 5
A turning point at (−1,  3) fixes the completed-square form straight away: f(x) = a(x+1)2 + 3. Now use the other point: f(1) = 4a + 3 = −1, so a = −1. Expanding, f(x) = −(x+1)2 + 3 = −x2 − 2x + 2. Checking against the options is quicker than expanding all six: only one has a turning point at x = −1 with value 3.