2025 NYJC Promo Q3

The diagram shows the graph of curve \(C_1\), with equation \(y=\mathrm{f}(x)\).

The curve intersects the \(x\)-axis at the point \(A(5,0)\) and has a turning point at \(B(12,4)\). The equations of the asymptotes are \(x=0\) and \(y=2\).

1. Sketch the graph of \(y=\mathrm{f}'(x)\), stating the equations of any asymptotes and the coordinates of any point(s) where the curve crosses the \(x\)-axes.[3]

Another curve \(C_2\) has equation \((x-12)^2+\dfrac{(y-k)^2}{9}=1\), where \(k>4\).

3. Describe a sequence of transformations that will transform the graph of \(C_2\) onto the graph of \(x^2+y^2=1\).[3]
4. Determine the range of values of \(k\) for which \(C_1\) and \(C_2\) do not intersect.[1]