Figure 1 shows the graph of $y = f(x)$, $-5 \leq x \leq 5$ - Edexcel - A-Level Maths Pure - Question 3 - 2006 - Paper 5

For $y = |f(x)|$, the parts of the graph where $f(x)$ is negative will be reflected above the x-axis. The maximum point $M(2, 4)$ stays at the same coordinates, while any segment of the graph that was below the x-axis will now be positive. It's important to identify where the original graph crosses the x-axis and reflect those segments accordingly.

The graph $y = f(|x|)$ means you will take the original function $f(x)$ for $x \geq 0$ and reflect that portion into the negative x-axis. The point $M(2, 4)$ remains unchanged at $M(2, 4)$, but the left side of the graph will mirror the right side. Ensure the correct intervals are reflected accurately.

For all sketches:

• In the graph of $y = f(x) + 3$, the maximum turning point is at $(2, 7)$.
• In the graph of $y = |f(x)|$, the maximum turning point remains at $(2, 4)$.
• In the graph of $y = f(|x|)$, the maximum turning point is also $(2, 4)$.