The diagram below shows a sketch of the graph of $y = \mathrm{f}(x)$. The curve passes through the

Question

The diagram below shows a sketch of the graph of $y = \mathrm{f}(x)$. The curve passes through the points with coordinates $(1, 0)$ and $(3, 0)$, and has turning points at $(1, 0)$ and $(4, 4)$. The asymptotes are $x = 0$, $x = 2$ and $y = 2$.

The diagram below shows a sketch of the graph of $y = \mathrm{f}(x)$. The curve passes through the points with coordinates $(1, 0)$ and $(3, 0)$, and has turning points at $(1, 0)$ and $(4, 4)$. The asymptotes are $x = 0$, $x = 2$ and $y = 2$.
1. Sketch on separate diagrams, the graphs of $y = \mathrm{f}'(x)$,[3]
2. Sketch on separate diagrams, the graphs of $y = \frac{1}{\mathrm{f}(x)}$,[3]

showing clearly the main features of the graphs.

Describe a sequence of transformations which transform the graph of $y = \frac{x^2 + 1}{x}$ to the graph of $y = \frac{2x^2 - 5x + 4}{x - 2}$.[3]

Accepted Answer

(a)(i) Graph shown, with vertical asymptotes $x=0$ and $x=2$, horizontal asymptote $y=0$, and zeros at $x=1$ and $x=4$ (a)(ii) Graph shown, with vertical asymptotes $x=1$ and $x=3$, horizontal asymptote $y=\frac12$, holes at $(0,0)$ and $(2,0)$, and turning point $(4,\frac14)$ (b) Translate $2$ units right, translate $\frac32$ units up, then stretch parallel to the $y$-axis by scale factor $2$

2025 RI P1 Q5

2025 RI P1 Q5

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