<p>The diagram below shows the graph of the function $y=\mathrm{f}\left( x \right)$, where</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)=\frac{a{{x}^{2}}+bx+c+2}{x+d}$ and $a$, $b$, $c$ and $d$ are integer constants.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1759222408207-jc1q7i9b48.png" width="400" class="aligncenter" /></figure>
<p>The curve has a vertical asymptote at the $y$-axis and an oblique asymptote which intersects the axes at $x=1$ and $y=-1$. Find the values of $a$, $b$ and $d$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">By considering$\frac{\mathrm{d}y}{\mathrm{d}x}$, show that $c<-2$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using the values of $a$, $b$ and $d$ found in (i), sketch ${{C}_{1}}$, the graph of $y=\mathrm{f}\left( x \right)$ if $c=-1$, labelling clearly the equation of asymptotes, coordinates of the turning points and axial intercepts (if any).</span></div></li></ol>
<p>On the same diagram, sketch the curve ${{C}_{2}}$, defined by the parametric equations</p>
<p style="text-align: center">$x=t-\cos t$, $y=2\sin t$, $0\le t\le \pi $.</p>
<p>Deduce the number of intersection points between ${{C}_{1}}$ and ${{C}_{2}}$.</p>

The diagram below shows the graph of the function $y=\mathrm{f}\left( x \right)$, where

$\mathrm{f}\left( x \right)=\frac{a{{x}^{2}}+bx+c+2}{x+d}$ and $a$, $b$, $c$ and $d$ are integer constants.

The curve has a vertical asymptote at the $y$-axis and an oblique asymptote which intersects the axes at $x=1$ and $y=-1$. Find the values of $a$, $b$ and $d$.

By considering$\frac{\mathrm{d}y}{\mathrm{d}x}$, show that $c<-2$.

Using the values of $a$, $b$ and $d$ found in (i), sketch ${{C}_{1}}$, the graph of $y=\mathrm{f}\left( x \right)$ if $c=-1$, labelling clearly the equation of asymptotes, coordinates of the turning points and axial intercepts (if any).

On the same diagram, sketch the curve ${{C}_{2}}$, defined by the parametric equations

$x=t-\cos t$, $y=2\sin t$, $0\le t\le \pi $.

Deduce the number of intersection points between ${{C}_{1}}$ and ${{C}_{2}}$.

Parametric Equations

Graphing Techniques

International Baccalaureate Higher Level Mathematics

NJC Graphing Techniques Tutorial Q4

NJC Graphing Techniques Tutorial Q4

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