Graphs & Transformations: Transformation to \(1/f(x)\) — HCI 2025 H2 Math Prelim Paper 1
What this question tests
Question
Given that \(b\) is a real constant such that \(0 < b < 4\), describe fully a sequence of transformations that transforms the curve \(y = x^2\) to the curve \(y = 4x^2 + bx + 1\).
Sketch the curve \(y = \dfrac{1}{4x^2 + bx + 1}\). Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of \(b\) where appropriate.
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Complete the square. \[\begin{aligned} y &= 4x^2 + bx + 1\\ &= 4\!\left(x^2 + \tfrac{b}{4}x + \tfrac{1}{4}\right)\\ &= 4\!\left[\left(x + \tfrac{b}{8}\right)^{\!2} + \tfrac{1}{4} - \tfrac{b^2}{64}\right]\\ &= 4\!\left(x + \tfrac{b}{8}\right)^{\!2} + 1 - \tfrac{b^2}{16}. \end{aligned}\]
Method 1 â translate, vertical scale, translate
(1) Translate the graph \(\dfrac{b}{8}\) units in the negative \(x\)-direction.
(2) Scale the graph parallel to the \(y\)-axis by a scale factor of \(4\).
(3) Translate the graph \(\left(1 - \dfrac{b^2}{16}\right)\) units in the positive \(y\)-direction.
Method 2 â translate, horizontal scale, translate
(I) Translate the graph \(\dfrac{b}{4}\) units in the negative \(x\)-direction.
(II) Scale the graph parallel to the \(x\)-axis by a scale factor of \(\dfrac{1}{2}\).
(III) Translate the graph \(\left(1 - \dfrac{b^2}{16}\right)\) units in the positive \(y\)-direction.
Sketch of \(y = \dfrac{1}{4x^2 + bx + 1}\).
\[ \left(-\dfrac{b}{8},\, \dfrac{16}{16 - b^2}\right). \]
{ \itshape (Sketched for \(b = 2\) as an illustration; topology is the same for \(0 < b < 4\).)