1. The problem asks to identify the graph of the function $y = \lfloor x \rfloor - 2$. 2. The floor function $\lfloor x \rfloor$ returns the greatest integer less than or equal to $x$. This means the graph is a step function with steps of length 1 on the x-axis. 3. Each step is horizontal and spans from an integer $n$ to $n+1$ on the x-axis, with the function value constant at $\lfloor x \rfloor - 2 = n - 2$ for $x$ in $[n, n+1)$. 4. Important rule: For the floor function, the left endpoint of each step is a closed circle (included), and the right endpoint is an open circle (excluded). 5. For example, at $x = -5$, $\lfloor -5 \rfloor = -5$, so $y = -5 - 2 = -7$. The step from $x = -5$ to $x = -4$ has $y = -7$ with a closed circle at $x = -5$ and open circle at $x = -4$. 6. The problem's first graph description matches this pattern: horizontal segments each 1 unit long, left end closed circle, right end open circle, left-most segment from $(-5, -5)$ to $(-4, -5)$, each segment 1 unit higher and 1 unit to the right than the previous. 7. However, since the function is $y = \lfloor x \rfloor - 2$, the y-values are shifted down by 2 compared to $\lfloor x \rfloor$. So at $x = -5$, $y = -7$, not $-5$. 8. The third graph description has segments from $(-3, -5)$ to $(-2, -5)$, which is shifted down by 2 compared to the floor function steps starting at $-3$. 9. The third graph also has left end closed circle and right end open circle, which matches the floor function behavior. 10. Therefore, the correct graph is the third one: step graph with horizontal segments 1 unit long, left end closed circle, right end open circle, left-most segment from $(-3, -5)$ to $(-2, -5)$, each segment 1 unit higher and 1 unit farther right, right-most segment from $(4, 2)$ to $(5, 2)$. Final answer: The graph described in the third option corresponds to $y = \lfloor x \rfloor - 2$.