<p>The curve with equation \(y = \mathrm{f}(x)\) for \(x \le b\), where \(\mathrm{f}(x)\) 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/1768123773445-zo22ymyd5pp.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Graph Sketching</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of \(y = \mathrm{f}(|x|)\), labelling the points where the graph intersects or touches the axes.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">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><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The function \(y = \mathrm{g}(x)\) is such that \(\mathrm{g}(x) = \mathrm{f}(x)\) for \(x \le k\), and \(y = \mathrm{g}^{-1}(x)\) exists.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the largest possible value of \(k\) in terms of \(a\) and \(b\).</div></li><li><div class="list-item-paragraph">Using the value of \(k\) found in (b)(i), 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 \(b\).</div></li><li><div class="list-item-paragraph">Explain why the solution to \(\mathrm{g}^{-1}(x) = x\) satisfies the equation \(\mathrm{g}^{-1}(x) = \mathrm{g}(x)\).</div></li></ol></li></ol>

The curve with equation \(y = \mathrm{f}(x)\) for \(x \le b\), 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.

Sketch the graph of \(y = \mathrm{f}(|x|)\), labelling the points where the graph intersects or touches the axes.

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.

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

Functions

International Baccalaureate Higher Level Mathematics

2025 ACJC P1 Q8

2025 ACJC P1 Q8