ASRJC Curve Sketching Tutorial T1-2 Q6

A sketch of the curve \({y = \frac{ax^2 + bx + c}{x + d}}\) where \(a\), \({a, b, c}\), \(c\) and \({d}\) are constants, is shown, not drawn to scale, in the diagram. The upper branch of the curve cuts the \(y-\)axis at the point \({A\left(0, \frac{3}{2}\right)}\) and the equation of the asymptotes, also shown in the diagram, are \({x = -2}\) and \({y = x - 3}\).

1. Write down the value of \({d}\) and determine the values of \(a\), \({a, b, c}\) and \(c\).[5]
2. Prove that \({y}\) cannot take any values in the interval \({(p, q)}\) where \({p}\) and \({q}\) are constants to be determined.[3]
3. Find the range of values of \({x}\) for which the curve is decreasing and concave downwards.[2]
4. Determine the range of values of \({m}\) if the equation \({ax^2 + bx + c = m(x + d)}\) has \(2\) negative roots.[1]