<p>The curve \(C\) has equation \(y = \frac{x^2 + ax + b}{x + 1}\), where&nbsp;\(a\) and \(b\) are constants. It is given that&nbsp;\(C\) has a minimum point at \((2,0)\).</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;\(a = -4\) and \(b = 4\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of \(C\), indicating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By adding an appropriate graph to your sketch in \(\mathbf{(ii)}\), deduce the number of real roots of the equation\n\[x^3 + ax^2 + (b + 1)x + 1 = 0.\]<span class="inline-marks">[2]</span></div></li></ol>

The curve \(C\) has equation \(y = \frac{x^2 + ax + b}{x + 1}\), where \(a\) and \(b\) are constants. It is given that \(C\) has a minimum point at \((2,0)\).

Sketch the graph of \(C\), indicating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts.

By adding an appropriate graph to your sketch in \(\mathbf{(ii)}\), deduce the number of real roots of the equation\n\[x^3 + ax^2 + (b + 1)x + 1 = 0.\]

Graphing Techniques

International Baccalaureate Higher Level Mathematics

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