Figure 1 shows a sketch of the graph of $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 2

For the graph of $y = f(x + 2) + 3$, we first translate the original graph to the left by 2 units. The maximum point $A(2, 3)$ translates to $A'(0, 3)$. Then, we raise the entire graph by 3 units, so the new maximum point becomes $A'(0, 6)$. The $y$-intercept shifts from $(0, 1)$ to $(0, 4)$.

To graph $y = 2f(2x)$, we first compress the graph horizontally by a factor of 2, affecting the maximum point. The new maximum from the original at $A(2, 3)$ moves to $A'(1, 3)$. Then, the graph is vertically stretched by a factor of 2, moving $A'$ to $A'(1, 6)$. The $y$-intercept also changes, scaling from $(0, 1)$ to $(0, 2)$.