The diagram below shows the graph of curve $C_1$, with equation $y = \mathrm{f}(x)$, defined for

Question

The diagram below shows the graph of curve $C_1$, with equation $y = \mathrm{f}(x)$, defined for $x \in \mathbb{R} \setminus \{4\}$, with asymptotes $x = 4$, and $y = -\frac{1}{2}x + 1$, and cuts the $x$-axis at the origin and $(6,0)$. Sketch the graph of $y = \mathrm{f}'(x)$, labelling the equations of asymptotes if any.[2] Given another curve $C_2$ has equation $\frac{(y+1)^2}{k^2} - (x-4)^2 = 1$, where $k$ is a positive constant. Describe a sequence of transformations that transform the graph of $y^2 - x^2 = 1$ onto the graph of $C_2$.[3] Determine the range of values of $k$ for which $C_1$ and $C_2$ intersect exactly $2$ times.[1]

Accepted Answer

(a) Asymptotes $x=4$, $y=-\frac12$; the derivative approaches the horizontal asymptote from below; see sketch. (b) Translate 4 units right, stretch by $k$ parallel to the $y$-axis, translate 1 unit down. (c) $k\geq\frac12$.

2024 ASRJC Promo Q6

2024 ASRJC Promo Q6

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