<p>The curve with equation \(y = \mathrm{f}(x)\) for&nbsp;\(x \leq b\), where&nbsp;\(\mathrm{f}(x)\) is a <strong>quadratic</strong> function, intersects the \(x\)-axis at the points \((a,0)\) and \((b,0)\), and the \(y\)-axis at the point \((0,c)\). The graph of \(y = \mathrm{f}(x)\) is shown in Figure 1. The scales on the \(x\)- and \(y\)-axes are the same.</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/1783531831399-rvzkp28l39.png" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Figure 1" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}(\left| x \right|)\), labelling the points where the graph intersects or touches the axes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Sketch the graph of \(y = \frac{1}{\mathrm{f}(x)}\), labelling the points where the graph intersects or touches the axes, as well as the equations of any asymptotes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-iii" data-list-item-path="item-a/item-a-iii"><div class="list-item-paragraph">Describe fully a sequence of transformations which transforms the graph of \(y = \mathrm{f}(2x+1)\) onto the graph of \(y = \mathrm{f}(x)\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The function \(y = \mathrm{g}(x)\) is such that&nbsp;\(\mathrm{g}(x) = \mathrm{f}(x)\) for&nbsp;\(x \leq k\), and \(y = \mathrm{g}^{-1}(x)\) exists.</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">Find the largest possible value of \(k\) in terms of \(a\) and \(b\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Using the value of \(k\) found in <strong>(b)(i)</strong>, sketch the graph of \(y = \mathrm{g}^{-1}(x)\) on Figure 1 in the Answer Booklet, labelling the intersections with the axes. Write down the range of \(\mathrm{g}^{-1}\) in terms of \(a\) and \(b\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-iii" data-list-item-path="item-b/item-b-iii"><div class="list-item-paragraph">Explain why the solution to \(\mathrm{g}^{-1}(x) = x\) satisfies the equation \(\mathrm{g}^{-1}(x) = \mathrm{g}(x)\).<span class="inline-marks">[1]</span></div></li></ol></li></ol>

The curve with equation \(y = \mathrm{f}(x)\) for \(x \leq b\), where \(\mathrm{f}(x)\) is a **quadratic** function, intersects the \(x\)-axis at the points \((a,0)\) and \((b,0)\), and the \(y\)-axis at the point \((0,c)\). The graph of \(y = \mathrm{f}(x)\) is shown in Figure 1. The scales on the \(x\)- and \(y\)-axes are the same.

The function \(y = \mathrm{g}(x)\) is such that \(\mathrm{g}(x) = \mathrm{f}(x)\) for \(x \leq k\), and \(y = \mathrm{g}^{-1}(x)\) exists.

2025 ACJC P1 Q8

2025 ACJC P1 Q8