June 2022 Paper 3 Q2
2
The function f is defined by \(\mathrm{f}(x) = x^3 - 8\).
| Scheme | Marks | AO |
|---|---|---|
| Translation | B1 | 2.5 |
| \(-8\) units parallel to the \(y\)-axis or \(\begin{pmatrix} 0 \\ -8 \end{pmatrix}\) | B1 | 1.1 |
| [2] |
Notes
B1: Do not accept shift, move, transformation, etc. for first B1
B1: Correct description e.g. correct vector (not as a coordinate), ‘8 units down’. Do not allow second B1 after incorrect type of transformation e.g. stretch/rotation etc. but allow after shift/move etc.
Condone lack of ‘units’ but do not accept ‘factor \(-8\)’, ‘8 places/spaces/steps down’ etc.
For ‘parallel to the \(y\) axis’ allow ‘vertically’, ‘in the \(y\) direction’. Do not accept across/up/along/to/in/towards the \(y\) axis’
If more than a single transformation, then no marks (unless two translations equivalent to the correct answer)
Mark vector before description/words
| Scheme | Marks | AO |
|---|---|---|
| \(y = x^3 - 8 \Rightarrow y + 8 = x^3\) \(x = \ldots\) | M1 | 1.1 |
| \(\mathrm{f}^{-1}(x) = (x + 8)^{\frac{1}{3}}\) | A1 | 1.1 |
| [2] |
Notes
M1: Attempt to make \(x\) the subject (allow sign errors only)
May use either \(\mathrm{f}(x)\) or \(y\)
A1: Must be in terms of \(x\). Allow the expression only e.g. \((x + 8)^{\frac{1}{3}}\), \(\sqrt[3]{x + 8}\), oe
Ignore what this expression is equated to
| Scheme | Marks | AO |
|---|---|---|
| One graph is the reflection of the other graph in the line \(y = x\) | B1 | 1.2 |
| [1] |
Notes
B1: Must include both ‘reflection’ or ‘mirror image’ or ‘mirrored’ and ‘\(y = x\)’
B0 for ‘symmetrical’ unless clearly describing reflective symmetry