Given that $b$ is a real constant such that $0

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 $y=4x^2+bx+1$.[4] Sketch the curve $y=\frac{1}{4x^2+bx+1}$. Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of $b$ where appropriate.[3]

Accepted Answer

Translation $b/8$ unit left; stretch parallel to the $y$-axis by factor $4$; translation $1-b^2/16$ unit up. Reciprocal: maximum $(-b/8,\ 16/(16-b^2))$, $y$-intercept $(0,1)$, horizontal asymptote $y=0$, no $x$-intercepts.

2025 HCI P1 Q6

2025 HCI P1 Q6

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