Problem

The curve $C$ with equation $y=f(x)$ is shown in Figure 4.

Figure 4

The curve $C$ has a single turning point, a maximum at $(4,9)$, crosses the coordinate axes at only two places, $(-3,0)$ and $(0,6)$, and has a single asymptote with equation $y=4$.

(a) State the equation of the asymptote to the curve with equation $y=f(-x)$.

(b) State the coordinates of the turning point on the curve with equation $y=f\left(\dfrac14x\right)$.

Given that the line with equation $y=k$, where $k$ is a constant, intersects $C$ at exactly one point,

(c) state the possible values for $k$.

The curve $C$ is transformed to a new curve that passes through the origin.

(d) (i) Given that the new curve has equation $y=f(x)-a$, state the value of the constant $a$.

(ii) Write down an equation for another single transformation of $C$ that also passes through the origin.