Higher June 2021 Paper 2 Q22
22 The curve S has equation \(y = \mathrm{f}(x)\) where \(\mathrm{f}(x) = x^2\)
The curve T has equation \(y = \mathrm{g}(x)\) where \(\mathrm{g}(x) = 2x^2 - 12x + 13\)
By writing \(\mathrm{g}(x)\) in the form \(a(x - b)^2 - c\), where \(a\), \(b\) and \(c\) are constants, describe fully a series of transformations that map the curve S onto the curve T.
(4)
| Scheme | Marks |
|---|---|
| \([\mathrm{g}(x) =]\ 2(x - 3)^2 - 5\) | B2 |
| stretch \(y\) direction scale factor 2 oe [ft their \(a\)] or translation \(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\) (ft correct use of their \(b\) and \(c\)) oe | M1 |
| Correct transformations in correct order | A1 |
| (4) | |
| (4 marks) |
Notes
B2: for \(a = 2\), \(b = 3\) and \(c = 5\) correct (stated or shown)
(B1 for one of \(a = 2\), \(b = 3\) and \(c = 5\) correct)
M1: Stretch and a correct description of the stretch or translation and a correct description of the translation
NB: must include the word translation (or translate) and stretch
A1: Stretch \(y\) direction scale factor 2 followed by translation \(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\) oe eg
translation \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\), stretch SF2 in \(y\) direction followed by translation \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\)
| Scheme | Marks |
|---|---|
| \([\mathrm{g}(x) =]\ 2(x - 3)^2 - 5\) | B2 |
| translation \(\begin{pmatrix} 3 \\ -2.5 \end{pmatrix}\) (ft correct use of their \(b\) and \(0.5c\)) oe or stretch \(y\) direction scale factor 2 (ft their \(a\)) | M1 |
| Correct transformations in correct order | A1 |
Notes
B2: for \(a = 2\), \(b = 3\) and \(c = 5\) correct (stated or shown)
(B1 for one of \(a = 2\), \(b = 3\) and \(c = 5\) correct)
M1: A correct description of the stretch or the translation
A1: Translation \(\begin{pmatrix} 3 \\ -2.5 \end{pmatrix}\) oe followed by stretch \(y\) direction scale factor 2