2026 RI P1 Q10

It is given that \(\mathrm f(x)=\frac{x^2-20x+109}{x-10}\). Describe the transformation which transforms the graph of \(y=x+\frac9x\) onto the graph of \(y=\mathrm f(x)\).[2]

Sketch the graph of \(y=\mathrm f(x)\), labelling clearly the equations of any asymptotes and the coordinates of any axial intercepts and turning points.[3]

On a separate diagram, sketch the graph of \(y=\mathrm f'(x)\), labelling clearly the equations of any asymptotes and the coordinates of any axial intercepts and turning points.[3]

By considering the graph of \((x-13)^2+\frac{y^2}{m}=1\), where \(m>0\), determine the value of \(m\) for which the equation \(m(x-13)^2+\left(\frac{x^2-20x+109}{x-10}\right)^2=m\) has exactly one real root.[2]