Which of the following functions describe the graph shown in the below figure?

The correct answer is: 𝑦 = ||𝑥| − 1| − 1

Step-by-Step Analysis:
By carefully examining the provided graph, we can extract key coordinate points $(x,y)$ through which the curve passes:
• At $x=0$, the graph passes through the origin, so $y=0$.
• At $x=1$ and $x=-1$, the graph has local minimum points at $y=-1$.
• At $x=2$ and $x=-2$, the graph intersects the x-axis, so $y=0$.
• At $x=3$ and $x=-3$, the graph reaches $y=1$.

Let's substitute these key x-values into our correct function $y=\left|\right|x|-1|-1$ to verify:
• For $x=0$:
$y=\left|\right|0|-1|-1=|0-1|-1=1-1=0$ (Matches the graph point (0, 0))
• For $x=\pm 1$:
$y=\left|\right|\pm 1|-1|-1=|1-1|-1=0-1=-1$ (Matches the graph points (-1, -1) and (1, -1))
• For $x=\pm 2$:
$y=\left|\right|\pm 2|-1|-1=|2-1|-1=1-1=0$ (Matches the graph points (-2, 0) and (2, 0))
• For $x=\pm 3$:
$y=\left|\right|\pm 3|-1|-1=|3-1|-1=2-1=1$ (Matches the graph points (-3, 1) and (3, 1))

Understanding via Graph Transformations:
1. Start with the standard absolute value function:
$y_1=\left|x\right|$
This is a standard V-shape with a vertex at $(0,0)$.
2. Shift the graph down by 1 unit:
$y_2=\left|x\right|-1$
The vertex moves down to $(0,-1)$, and the x-intercepts are at $x=-1$ and $x=1$.
3. Take the absolute value of the entire function:
$y_3=\left|\right|x|-1|$
This reflects any negative y-values (the section between $x=-1$ and $x=1$ where the vertex was at $(0,-1)$) over the x-axis, changing the vertex to $(0,1)$ and creating a W-shaped graph.
4. Shift the W-shaped graph down by 1 unit:
$y=\left|\right|x|-1|-1$
This shifts the entire W-shape downwards, moving the corner points to $(-1,-1)$, $(0,0)$, and $(1,-1)$, which corresponds exactly to the given plot.