Graphs & Transformations: f(a-x), \(|\)g(x)\(|\) & g(\(|\)x\(|\)) — NYJC 2025 H2 Math Prelim Paper 1
What this question tests
Question
(a) The graph of \(y = \mathrm{f}(x)\) has turning points at \(A(-a,0)\), \(B(a,b)\) and \(C(c,-b)\) and intersects the \(x\)-axis at the points \(D(d,0)\) and \(E(e,0)\). The graph of \(y = \mathrm{f}(x)\) is as shown in the diagram below.
Sketch the graph of \(y = \mathrm{f}(a-x)\), labelling the coordinates of the corresponding points of \(A\), \(B\), \(C\), \(D\), and \(E\) clearly.
(b) The graph of \(y = \mathrm{g}(x)\) has asymptotes \(y=0\), \(y=-m\) and \(x=-n\), and intersects the \(y\)-axis at \(P(0,p)\), where \(p<m\). The graph of \(y=\mathrm{g}(x)\) is as shown in the diagram below.
Sketch on separate diagrams the graphs of
(i) \(y = |\mathrm{g}(x)|\) and
(ii) \(y = \mathrm{g}(|x|)\),
labelling clearly the asymptote(s) and coordinates of \(P\) on each graph.
Show full worked solution▾
(a) The transformation \(y = \mathrm{f}(a-x)\) is a reflection in the line \(x = \dfrac{a}{2}\), followed by a horizontal translation of \(a\) units. Equivalently, it is a reflection in \(x = \dfrac{a}{2}\). The coordinates transform as: \(x \mapsto a - x\).
The key point mappings are: \(A(-a,0)\to A'(2a,0)\), \(B(a,b)\to B'(0,b)\), \(C(c,-b)\to C'(a-c,-b)\), \(D(d,0)\to D'(a-d,0)\), \(E(e,0)\to E'(a-e,0)\).
(b)(i) \(y = |\mathrm{g}(x)|\): reflect the portion of the curve below the \(x\)-axis upward. The asymptote \(y=-m\) becomes \(y=m\); \(y=0\) remains; \(x=-n\) remains. Point \(P(0,p)\) stays.
(b)(ii) \(y = \mathrm{g}(|x|)\): keep the right-half curve (\(x \ge 0\)) and reflect it in the \(y\)-axis. The vertical asymptote \(x = -n\) disappears (and \(x=n\) is introduced by symmetry). Only \(y=0\) remains as a horizontal asymptote. \(P(0,p)\) stays.
