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 7 - 2010 - Paper 2

To sketch the graph of the function $y = ext{ln} |x|$, we first need to determine its key characteristics.

1. Domain: The function is defined for all real numbers except $x = 0$. This means the domain is (- ext{∞}, 0) igcup (0, ext{∞}).

2. Intercepts: The graph intersects the y-axis at the point $(0, - ext{∞})$, which we note but cannot explicitly mark since the function is undefined at $x = 0$.

   • For the x-intercept, we set $y = 0$: $0 = ext{ln} |x|$ This occurs when $|x| = 1$, therefore, the x-intercepts are at the coordinates $(-1, 0)$ and $(1, 0)$.

3. Behavior of the graph:

   • As $x$ approaches $0$ from the left (negative side), $y o - ext{∞}$.
   • As $x$ approaches $0$ from the right (positive side), $y o - ext{∞}$ as well.
   • For positive values of $x$, the graph rises without bound as $x$ increases.
   • For negative values of $x$, the graph mirrors the behavior of the positive side due to the absolute value in the logarithm function.

4. Shape: The shape of the graph consists of two branches:

   • A right-hand branch in quadrant 1 and quadrant 4.
   • A left-hand branch in quadrant 2 and quadrant 3, both curving upward as $|x|$ increases.

5. Final sketch: On the coordinate system, sketch the two branches of the function with the identified intercept points marked. Ensure the branches are correctly positioned in their respective quadrants and connected smoothly.