2017 H2 Mathematics Paper 1 Question 4

Graphs and Transformations

Answers

(ii)

{x=-2}

{y=4}

(iii)

Translate the graph of ${C}$ ${2}$ units in the positive ${x\textrm{-}}$ axis direction.
Translate the resulting graph ${4}$ units in the negative ${y\textrm{-}}$ axis direction.

Full solutions

(i)

\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{4(x+2)-1(4x+9)}{(x+2)^2} \\ &= \frac{- 1}{(x+2)^2} \end{align*}

Since

{(x+2)^2 > 0}

for all

{x \in \mathbb{R}, x \neq -2, }

\frac{\mathrm{d}y}{\mathrm{d}x} < 0

Hence the gradient of

{C}

is negative for all points on

{C \; \blacksquare}

(ii)

\begin{align*} y &= \frac{4x+9}{x+2} \\ &= \frac{4x+8+1}{x+2} \\ &= 4 + \frac{1}{x+2} \end{align*}

Hence the equations of the asymptotes of

{C:}

\begin{align*} x &= -2 \; \blacksquare \\ y &= 4 \; \blacksquare \end{align*}

(iii)

Translate the graph of ${C}$ ${2}$ units in the positive ${x\textrm{-}}$ axis direction.
Translate the resulting graph ${4}$ units in the negative ${y\textrm{-}}$ axis direction.

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