Sketch the graph of $y = \ln |x|$, stating the coordinates of any points of intersection with the axes. - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 2

To sketch the graph of the function $y = \ln |x|$, we first need to understand its properties.

1. Identifying the Domain: The function is defined for all $x \neq 0$. Hence, the domain is $(-\infty, 0) \cup (0, \infty)$.

2. Finding Points of Intersection with the Axes:

   • The graph intersects the y-axis when $x = 1$ and $y = \ln |1| = \ln 1 = 0$. Therefore, one point of intersection is (1, 0).
   • As $x$ approaches 0 from the right, $\ln |x| \to -\infty$, and as $x$ approaches 0 from the left, the same occurs due to the absolute value.

3. Graph Shape:

   • For $x > 0$: The graph will have a right-hand branch in Quadrant I, starting at (1, 0) and approaching $(-\infty)$ as $x \to 0^+$.
   • For $x < 0$: The graph mirrors itself, showing a left-hand branch in Quadrant II, also approaching $(-\infty)$ as $x \to 0^-$. It intersects the x-axis at (-1, 0).

4. Final Touch: The graph is symmetric, clearly demonstrating the behavior of the logarithmic function around the y-axis. The correct points of intersection are therefore stated: (-1, 0) and (1, 0).