Standard +0.3 This is a straightforward application of function transformations in sequence. Students must systematically apply three standard transformations to y = ln x (reflection, stretch, translation) and simplify to the required form. While it requires careful tracking of multiple steps, each transformation is routine and the algebraic manipulation uses basic log laws. Slightly easier than average as it's methodical rather than requiring insight.
2 The curve \(y = \ln x\) is transformed by:
a reflection in the \(x\)-axis, followed by a stretch with scale factor 3 parallel to the \(y\)-axis, followed by a translation in the positive \(y\)-direction by \(\ln 4\).
Find the equation of the resulting curve, giving your answer in the form \(y = \ln ( \mathrm { f } ( x ) )\).
Apply one of the transformations correctly to their equation
B1
Obtain correct \(-3\ln x + \ln 4\)
B1
or equiv
Show at least one logarithm property
M1
correctly applied to their equation of resulting curve (even if errors have been made earlier)
Obtain \(y = \ln(4x^{-3})\)
A1
4 marks: or equiv of required form; \(\ln 4x^{-3}\) earns A1; correct answer only earns 4/4; condone absence of \(y =\)
Apply one of the transformations correctly to their equation | B1 |
Obtain correct $-3\ln x + \ln 4$ | B1 | or equiv
Show at least one logarithm property | M1 | correctly applied to their equation of resulting curve (even if errors have been made earlier)
Obtain $y = \ln(4x^{-3})$ | A1 | 4 marks: or equiv of required form; $\ln 4x^{-3}$ earns A1; correct answer only earns 4/4; condone absence of $y =$
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2 The curve $y = \ln x$ is transformed by:\\
a reflection in the $x$-axis, followed by a stretch with scale factor 3 parallel to the $y$-axis, followed by a translation in the positive $y$-direction by $\ln 4$.\\
Find the equation of the resulting curve, giving your answer in the form $y = \ln ( \mathrm { f } ( x ) )$.
\hfill \mbox{\textit{OCR C3 2011 Q2 [4]}}