Figure 1 shows part of the curve with equation $y = f(x)$, $x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 25 - 2013 - Paper 1

To sketch the inverse function $y = f^{-1}(x)$, we reflect the original function $y = f(x)$ over the line $y = x$. The critical points are approximately $(0, 2)$ and $(2, 0)$, leading to the points:

• $(2, 0)$
• $(0, 2)$

This indicates that the inverse graph increases from $(-3, 0)$ upwards, maintaining the shape of the original graph.

To sketch the curve $y = f(x) - 2$, we take the original function and shift it down by 2 units. The key points will shift accordingly:

• Original point $Q(0, 2)$ moves to $(0, 0)$.
• Original point $P(-3, 0)$ moves to $(-3, -2)$.

This transformation maintains the shape, illustrating a downward shift.

For the equation $y = 2f(\frac{1}{2} x)$, we first horizontally stretch the graph by a factor of 2 and then vertically stretch by a factor of 2. The significant points change as follows:

• The point $(0, 2)$ on the original curve will transform to $(0, 4)$.
• The point $(-3, 0)$ will transform to $(-6, 0)$.

Sketching these will result in a more pronounced upward curve, with specific points reflecting the transformations.