Problem

Figure 2 shows a sketch of part of the curve with equation $y=f(x)$.

Figure 2

The curve crosses the $y$-axis at the point $(0,8)$.

The line with equation $y=10$ is the only asymptote to the curve.

The curve has a single turning point, a minimum point at $(2,5)$, as shown in Figure 2.

(a) State the coordinates of the minimum point of the curve with equation

$$\begin{align*} y=f\left(\frac14x\right) \end{align*}$$

(b) State the equation of the asymptote to the curve with equation $y=f(x)-3$

The curve with equation $y=f(x)$ meets the line with equation $y=k$, where $k$ is a constant, at two distinct points.

(c) State the set of possible values for $k$.

(d) Sketch the curve with equation $y=-f(x)$. On your sketch, show clearly the coordinates of the turning point, the coordinates of the intersection with the $y$-axis and the equation of the asymptote.