Graphs & Transformations: Translate, reciprocal & modulus — EJC 2025 H2 Math Prelim Paper 1
What this question tests
Question
The diagram below shows the graph of \(y = \mathrm{f}(2x)\). The graph has a turning point at \((2,0)\), and asymptotes with equations \(x = 0\) and \(y = k\).
- State a single transformation that will transform the graph of \(y = \mathrm{f}(2x)\) onto the graph of \(y = \mathrm{f}(2x+4)\). Hence sketch the graph of \(y = \mathrm{f}(2x+4)\).
On separate clearly labelled diagrams, sketch the graphs of
- \(y = \dfrac{1}{\mathrm{f}(2x)}\),
- \(y = -\mathrm{f}(|x|)\).
Show full worked solution▾
Write \(\mathrm{f}(2x+4) = \mathrm{f}(2(x+2))\). This is a translation of the graph of \(y=\mathrm{f}(2x)\) by \(2\) units in the negative \(x\)-direction.
The translated graph has: turning point at \((0,0)\), vertical asymptote \(x=-2\), horizontal asymptote \(y=k\).
Translation of 2 units in the negative \(x\)-direction.
Key features: where \(\mathrm{f}(2x) \to \infty\), the reciprocal \(\to 0\); the zero of \(\mathrm{f}(2x)\) at \((2,0)\) becomes a vertical asymptote at \(x=2\); horizontal asymptote becomes \(y = \tfrac{1}{k}\).
