June 2022 Paper 2 Q8
8
(a) Sketch the graph of \(y = \dfrac{1}{x^2}\) [2 marks]
(b) The graph of \(y = \dfrac{1}{x^2}\) can be transformed onto the graph of \(y = \dfrac{9}{x^2}\) using a stretch in one direction.
Beth thinks the stretch should be in the \(y\)-direction.
Paul thinks the stretch should be in the \(x\)-direction.
State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Sketches a branch of the curve with correct shape in 1st or 2nd quadrant with correct asymptotes and not touching the axes | M1 | 1.2 |
| Sketches a fully correct curve with correct asymptotes and not touching the axes | A1 | 1.1b |
| (2) |
Typical solution
| Scheme | Marks | AO |
|---|---|---|
| Describes a stretch of scale factor 3 or 9 in either direction | M1 | 2.4 |
| Explains that the curve can be stretched in the \(y\)-direction by a scale factor of 9 Scale factor PI by \(9\mathrm{f}(x)\) OR Explains that the curve can be stretched in the \(x\)-direction by a scale factor of 3 Scale factor PI by \(\mathrm{f}\left(\dfrac{x}{3}\right)\) | A1 | 2.4 |
| Explains that the curve can be stretched in the \(y\)-direction by a scale factor of 9 PI by \(9\mathrm{f}(x)\) AND Explains that the curve can be stretched in the \(x\)-direction by a scale factor of 3 PI by \(\mathrm{f}\left(\dfrac{x}{3}\right)\) AND Concludes both are correct | R1 | 2.2a |
| (3) | ||
| (5 marks) |
Typical solution
The curve can be stretched in the \(y\)-direction by a scale factor of 9
The curve can be stretched in the \(x\)-direction by a scale factor of 3
Beth and Paul are both correct.