<p>The diagram below shows the graph of curve \(C_1\), with equation \(y = \mathrm{f}(x)\), defined for&nbsp;\(x \in \mathbb{R} \setminus \{4\}\), with asymptotes \(x = 4\), and \(y = -\frac{1}{2}x + 1\), and cuts the \(x\)-axis at the origin and \((6,0)\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1779426857358-vp17jeaoln.png" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Graph of curve C1" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}'(x)\), labelling the equations of asymptotes if any.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given another curve \(C_2\) has equation \(\frac{(y+1)^2}{k^2} - (x-4)^2 = 1\), where&nbsp;\(k\) is a positive constant. Describe a sequence of transformations that transform the graph of \(y^2 - x^2 = 1\) onto the graph of \(C_2\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Determine the range of values of \(k\) for which \(C_1\) and \(C_2\) intersect exactly \(2\) times.<span class="inline-marks">[1]</span></div></li></ol>

The diagram below shows the graph of curve \(C_1\), with equation \(y = \mathrm{f}(x)\), defined for \(x \in \mathbb{R} \setminus \{4\}\), with asymptotes \(x = 4\), and \(y = -\frac{1}{2}x + 1\), and cuts the \(x\)-axis at the origin and \((6,0)\).

Sketch the graph of \(y = \mathrm{f}'(x)\), labelling the equations of asymptotes if any.

Given another curve \(C_2\) has equation \(\frac{(y+1)^2}{k^2} - (x-4)^2 = 1\), where \(k\) is a positive constant. Describe a sequence of transformations that transform the graph of \(y^2 - x^2 = 1\) onto the graph of \(C_2\).

Determine the range of values of \(k\) for which \(C_1\) and \(C_2\) intersect exactly \(2\) times.

Transformations of Graphs

International Baccalaureate Higher Level Mathematics

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