The graph of $y=\mathrm{h}\left( x \right)$, where \(\mathrm{h}\left( x \right)={{\mathrm{e}}

Question

The graph of $y=\mathrm{h}\left( x \right)$, where $\mathrm{h}\left( x \right)={{\mathrm{e}}^{3x}}$, was transformed to a new graph with equation $y=\mathrm{g}\left( x \right)$. Given that $\mathrm{g}\left( x \right)=\frac{1}{\mathrm{h}'\left( x \right)}$, express $\mathrm{g}\left( x \right)$ in the form of $k{{\mathrm{e}}^{mx}}$ where $k$ and $m$ are exact real constants.[1] Describe a sequence of two transformations which would transform the graph of $y=\mathrm{h}\left( x \right)$ to the graph of $y=\mathrm{g}\left( x \right)$.[2] The diagram below shows the curve $y=\mathrm{f}\left( x \right)$. The curve has a minimum point $\left( -3,-4 \right)$ and passes through the origin. The lines $x=3$ and $y=-2$ are the vertical and horizontal asymptotes to the curve respectively. Sketch, including the coordinates of the point(s) of intersections with the axes, turning point(s) and equation(s), if any, the following: (a) $\quad y=-\frac{1}{2}\mathrm{f}\left( x-3 \right)$.[4] (b) $\quad y=\frac{1}{\mathrm{f}\left( x \right)}$.[3] State the coordinates of the point(s) where the curve $y=\mathrm{f}'\left( x \right)$ cuts the axes.[1]

Accepted Answer

(i) $	ext{g}\left( x ight)=\frac{1}{3}{{	ext{e}}^{-3x}}$ (ii) Reflection about the $y$-axis, A scaling parallel to the $y$-axis with a scale factor of $\frac{1}{3}$.

2021 SAJC Promo Q6

2021 SAJC Promo Q6

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