Sketch the graph of $y = ext{ln} |x|$, stating the coordinates of any points of intersection with the axes. - Edexcel - A-Level Maths Pure - Question 6 - 2010 - Paper 2

To sketch the graph of the function $y = ext{ln} |x|$, we need to consider the behavior of the function in different quadrants and its points of intersection with the axes.

1. Identify Points of Intersection with the Axes:

   • The graph intersects the x-axis where $y = 0$. This occurs when:

     $ext{ln} |x| = 0 \Rightarrow |x| = 1 \Rightarrow x = 1 \text{ or } x = -1$

   • Therefore, the points of intersection with the x-axis are at $(1, 0)$ and $(-1, 0)$.

2. Analyze the Behavior in Quadrants:
   • For $x > 0$, $y = ext{ln}(x)$ is defined, and it is a smooth curve that approaches negative infinity as $x$ approaches 0 from the right.
   • For $x < 0$, $y = ext{ln}(-x)$ is also defined, reflecting the graph from the first quadrant to the second quadrant, showing similar behavior as $x$ moves away from zero.
3. Sketch the Graph:
   • In quadrant 1, the graph rises above the x-axis with a vertical asymptote at $x=0$.
   • In quadrant 2, the graph reflects across the y-axis, maintaining the same shape, and continues upwards. In quadrants 3 and 4, the graph is mirrored similarly, creating a 'V' shape.

Overall, the finished graph should show two branches: one in quadrants 1 and 4 (for $x > 0$) and the other in quadrants 2 and 3 (for $x < 0$).

4. Final Result:
   • The coordinates of the points of intersection with the x-axis are $(-1, 0)$ and $(1, 0)$. The graph exhibits the characteristics discussed, capturing the correct shape and intersections.