Graphs and transformations – Year 12 A level maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   Each factor gives a root; the squared factor \((x-1)^2\) gives a repeated root where the curve touches the axis.

   Roots at \(x=-2\) (crosses) and \(x=1\) (touches, repeated) (2). \(y\)-intercept: \(1\times2=2\) at \((0,2)\) (1). Positive cubic: falls on the left, rises on the right, with a turning point on the axis at \((1,0)\) (1).

   So the curve crosses the axis at \((-2,0)\), meets the \(y\)-axis at \((0,2)\) and touches the \(x\)-axis at \((1,0)\).

2. Answer to Question 2 (4 marks)

   1. \(f(x+1)\) moves the graph 1 unit left. \((2,-2)\) (1)
   2. \(3f(x)\) multiplies \(y\) by 3. \((3,-6)\) (1)
   3. \(f(x/3)\) multiplies \(x\) by 3. \((9,-2)\) (1)
   4. Reflect in the \(x\)-axis first, so \(-2\) becomes 2, then add 2. \((3,4)\) (1: reflect to \(2\) then add 2) So \(P\) maps to \((3,4)\).

   In (c), the stretch with scale factor 3 parallel to the \(x\)-axis leaves \(y=-2\) unchanged. In (d), doing the translation before the reflection would give \((3,0)\), which is wrong.

3. Answer to Question 3 (4 marks)

   The graph is \(y=\dfrac1x\) translated by \(\begin{pmatrix}-2\\-3\end{pmatrix}\), so the asymptotes move to \(x=-2\) and \(y=-3\).

   Asymptotes \(x=-2\) and \(y=-3\) (2). \(y\)-intercept \(\tfrac12-3=-\tfrac52\) (1). \(x\)-intercept: \(\dfrac1{x+2}=3\), \(x=-\tfrac53\) (1).

   So the curve crosses the axes at \((0,-\tfrac52)\) and \((-\tfrac53,0)\).

   Sketch: the curve has two branches. For \(x\gt-2\) the branch lies above \(y=-3\) and passes through both intercepts; for \(x\lt-2\) it lies below \(y=-3\). Marks: 1 for each asymptote correctly labelled, 1 for each intercept. Check: at \(x=-\tfrac53\), \(\dfrac1{1/3}-3=0\) ✓.

4. Answer to Question 4 (3 marks)

   Inside the bracket \(x-5\) moves the graph 5 right; outside, \(+1\) moves it up. Replacing \(x\) by \(2x\) halves every \(x\)-coordinate.

   (a) Translation by \(\begin{pmatrix}5\\1\end{pmatrix}\) (2). (b) Stretch with scale factor \(\tfrac12\) parallel to the \(x\)-axis (1).

   A translation needs its vector; a stretch needs its scale factor and direction.

   Writing ‘move 5 right and 1 up’ scores the marks only if the word translation or the vector is given. In (b), a stretch of scale factor 2 is the most common wrong answer.

5. Answer to Question 5 (5 marks)

   1. A positive cubic with a repeated root at 0 and a single root at 3. Roots at \(0\) (touching) and \(3\); max \((0,0)\); min \((2,-4)\) (2)
   2. Count the intersections of the line \(y=k\) with the curve, using the turning points. \(k=-2\): 3 solutions (1). \(k=-4\): 2 solutions (1) For \(k=-4\) the line touches the minimum, giving a repeated root.
   3. Expand \((x-2)^2\) first, then multiply by \((x+1)\). \((x+1)(x^2-4x+4)=x^3-3x^2+4\) (1) This confirms the result.

Total : 20 marks