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

The function $y = ext{ln} |x|$ is defined for all $x eq 0$ and has two branches: one for positive $x$ and another for negative $x$.

1. Identify Points of Intersection with the Axes:
   • The graph intersects the y-axis when $x = 0$, but since the function is undefined at this point, there are no y-intercepts.
   • For x-intercepts, set $y = 0$:
   $$$$

ightarrow |x| = e^0 = 1$$
This means the graph intersects the x-axis at the points $(1, 0)$ and $(-1, 0)$.

2. Shape of the Graph:

   • For $x > 0$, the graph increases without bound as $x$ approaches infinity. The curve passes through (1, 0) and approaches the y-axis from the right, but never touches it.
   • For $x < 0$, the graph mirrors the right-hand side across the y-axis due to the absolute value. This means it is also increasing and passes through (-1, 0), approaching the y-axis from the left.

3. Sketching the Graph:

   • The graph will have a 'V' shape, with the vertex at the origin, as it curves upwards into quadrants I and II on the left and quadrants III and IV on the right, clearly indicating the expected shape outlined in the marking scheme.