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TMUA 2017 · Paper 2 · Question 14 of 20

TMUA 2017 Paper 2 Question 14

Graphs and transformations — Quadratics · how the vertex moves. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Graphs and transformationsQuadratics · how the vertex moves8 options

The original question includes a diagram: An upward parabola with vertex P above the x-axis and right of the y-axis, dashed lines joining P to both axes; the eight answer options repeat this picture with the new vertex placed in eight different positions relative to P.

The diagram shows the graph of y = x2 − 2bx + c, an upward parabola whose vertex P lies in the first quadrant: P is to the right of the y-axis and above the x-axis, with dashed lines drawn from P to each axis.

Which one of the following could be the graph of y = x2 − 2Bx + c, where B > b?

Each option shows an upward parabola drawn on the same axes, with the position of the original vertex P still marked by its dashed lines. The options differ only in where the new vertex sits relative to P.

  1. AThe new vertex is to the right of P and at the same height as P.
  2. BThe new vertex is to the left of P and at the same height as P.
  3. CThe new vertex is directly above P.
  4. DThe new vertex is directly below P.
  5. EThe new vertex is above P and to the right of it.
  6. FThe new vertex is below P and to the right of it.
  7. GThe new vertex is above P and to the left of it.
  8. HThe new vertex is below P and to the left of it.
Show the answer and worked solution
answer · F
  1. AThe new vertex is to the right of P and at the same height as P.
  2. BThe new vertex is to the left of P and at the same height as P.
  3. CThe new vertex is directly above P.
  4. DThe new vertex is directly below P.
  5. EThe new vertex is above P and to the right of it.
  6. FThe new vertex is below P and to the right of it.
  7. GThe new vertex is above P and to the left of it.
  8. HThe new vertex is below P and to the left of it.
Complete the square: y = (x−b)2 + c − b2, so P = (b,  c−b2), and the new curve has vertex (B,  c−B2). Since P is to the right of the y-axis, b > 0, and B > b > 0 moves the vertex to the right. The same inequality gives B2 > b2, so c − B2 < c − b2 and the vertex also moves down. Both changes happen together, which rules out every option that keeps the vertex level with P, directly above or below it, or to the left.