<ol class="abc-list"><li><div class="list-item-paragraph">The diagram shows the curve \(y = \mathrm{f}(x)\) with a maximum point at \(C(4,3)\). The curve crosses the axes at the points \(A(0,2)\) and \(B(3,0)\). The lines \(x = 2\) and \(y = 0\) are the asymptotes of the curve.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1779689201459-cxim9fz0yd5.png" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}'(x)\), clearly stating the equations of the asymptotes and the coordinates of the points corresponding to \(A\), \(B\) and \(C\) where appropriate.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The curve \(C_1\) has equation \(y = \frac{ax^2 + bx - 8}{x - 2}\), where&nbsp;\(a\) and \(b\) are constants. It is given that&nbsp;\(C_1\) has an asymptote \(y = 3 - 2x\).</div><ol class="i-list"><li><div class="list-item-paragraph">State the value of \(a\) and show that&nbsp;\(b = 7\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch \(C_1\), clearly stating the equations of any asymptotes, the coordinates of any stationary points and of any points where&nbsp;\(C_1\) crosses the axes.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The curve \(C_1\) is transformed by a translation of \(2\) units in the negative \(x\)-direction, followed by a stretch with scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis, to form the curve \(C_2\). Find the equation of \(C_2\).<span class="inline-marks">[2]</span></div></li></ol></li></ol>

The diagram shows the curve \(y = \mathrm{f}(x)\) with a maximum point at \(C(4,3)\). The curve crosses the axes at the points \(A(0,2)\) and \(B(3,0)\). The lines \(x = 2\) and \(y = 0\) are the asymptotes of the curve.

Sketch the graph of \(y = \mathrm{f}'(x)\), clearly stating the equations of the asymptotes and the coordinates of the points corresponding to \(A\), \(B\) and \(C\) where appropriate.

The curve \(C_1\) has equation \(y = \frac{ax^2 + bx - 8}{x - 2}\), where \(a\) and \(b\) are constants. It is given that \(C_1\) has an asymptote \(y = 3 - 2x\).

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