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

The correct answer is:
$$y=\left|\right|x|-1|-1$$

By analyzing the provided graph, we can identify several key coordinates and features:
1. The local minimums (bottom peaks/cusps) are located at $(-1,-1)$ and $(1,-1)$.
2. The local maximum is at the origin $(0,0)$.
3. The graph crosses the x-axis (x-intercepts) at $(-2,0)$ and $(2,0)$.
4. The graph passes through the points $(-3,1)$ and $(3,1)$.

We can verify the correct function by substituting these x-values:
• For $x=0$:
$$y=\left|\right|0|-1|-1=|0-1|-1=1-1=0$$
This correctly corresponds to the point $(0,0)$.
• For $x=\pm 1$:
$$y=\left|\right|\pm 1|-1|-1=|1-1|-1=0-1=-1$$
This correctly corresponds to the vertices at $(-1,-1)$ and $(1,-1)$.
• For $x=\pm 2$:
$$y=\left|\right|\pm 2|-1|-1=|2-1|-1=1-1=0$$
This correctly corresponds to the x-intercepts at $(-2,0)$ and $(2,0)$.

Alternatively, we can understand the graph step-by-step through transformations:
1. Start with the parent absolute value function: $y_1=\left|x\right|$ (V-shape with vertex at $(0,0)$).
2. Shift down by 1 unit: $y_2=\left|x\right|-1$ (vertex shifts to $(0,-1)$).
3. Reflect negative parts: Applying the outer absolute value $y_3=\left|\right|x|-1|$ reflects the portion below the x-axis upwards. The vertex at $(0,-1)$ reflects to $(0,1)$, and new cusps are formed on the x-axis at $(-1,0)$ and $(1,0)$.
4. Final Shift down by 1 unit: Subtracting 1 to get $y=\left|\right|x|-1|-1$ shifts the entire graph down. The peak at $(0,1)$ shifts to $(0,0)$ and the cusps shift down to $(-1,-1)$ and $(1,-1)$, perfectly matching the given graph.