Moderate -0.3 This is a multi-part C2 question covering standard transformations and exponential equations. Part (i) requires recognizing that (1/3)^x = 3^(-x) (reflection in y-axis). Part (ii) is routine sketching. Part (iii) involves solving 3^(-x) = 2(3^x), which reduces to a simple quadratic in 3^x, then using logarithms—all standard techniques with clear guidance from the question structure. Slightly easier than average due to the scaffolding and routine nature of each component.
5. (i) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(ii) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes.
The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(iii) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).
5. (i) Describe fully a single transformation that maps the graph of $y = 3 ^ { x }$ onto the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$.\\
(ii) Sketch on the same diagram the curves $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ and $y = 2 \left( 3 ^ { x } \right)$, showing the coordinates of any points where each curve crosses the coordinate axes.
The curves $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ and $y = 2 \left( 3 ^ { x } \right)$ intersect at the point $P$.\\
(iii) Find the $x$-coordinate of $P$ to 2 decimal places and show that the $y$-coordinate of $P$ is $\sqrt { 2 }$.\\
\hfill \mbox{\textit{OCR C2 Q5 [9]}}