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

To sketch the graph of $y = |f(x)|$, we reflect any portions of the graph of $f(x)$ that are below the x-axis across the x-axis.

From the original graph, if there are points where $f(x)$ is negative, those will now be positive in the $|f(x)|$ graph, while positive points remain unchanged. The turning points will be re-evaluated to show:

• The coordinates of the new maximum turning point will stay the same if they are above the x-axis; otherwise, they will change based on reflections.

To sketch the graph of $y = f(|x|)$, we consider the behavior of $f(x)$ for $x < 0$. Since $|x|$ gives the same value for both positive and negative x, we take the graph of $f(x)$ for positive values and mirror it for the negative side.

Hence, the graph will be symmetric about the y-axis. The maximum turning point M(2, 4) will be represented at M(-2, 4) as well, along with M(2, 4).