The function $y=\mathrm{f}\left( x \right)$ is defined by $y=\frac{{{x}^{2}}+x+p}{x+2}$, where&n

Question

The function $y=\mathrm{f}\left( x \right)$ is defined by $y=\frac{{{x}^{2}}+x+p}{x+2}$, where $p$ is a constant. Find the value of $p$ if $\mathrm{f}$ is a linear function.[1] If $p$ is greater than the value found in part (i), use an algebraic method to determine the range of values that $y$ can take, leaving your answer in terms of $p$.[3] Take $p=12$ for the rest of this question. Sketch the graph of $y=\frac{{{x}^{2}}+x+12}{x+2}$. Give the equation(s) of any asymptotes, the coordinates of any turning point(s) and the points of intersection with the axes.[3] Find the number of real roots of the equation ${{\left( {{x}^{2}}+x+12 \right)}^{2}}-{{\left( x+2 \right)}^{2}}\left( 36-{{x}^{2}} \right)=0$ by adding a suitable graph on the sketch in part (iii).[2] Describe a sequence of $3$ transformations which transforms the graph of $y=\frac{{{x}^{2}}+x+12}{x+2}$ onto $y=-2x+3-\frac{7}{x+1}$.[3]

Accepted Answer

(i) $p=-2$(ii) $y\ge 3+2\sqrt{2+p}$ or $y\le 3-2\sqrt{2+p}$ (iv) $2$

2022 HCI J1 BT Q8

2022 HCI J1 BT Q8

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