Question

answer
○ x
○ x+2
○ x-2
○ |x|+2
graph of a step - like function on a grid with marked points (filled and open circles) and axes with integer labels

Explanation:

Step1: Analyze the graph's transformation

The graph appears to be a transformation of the greatest integer function \( y = \lfloor x
floor \). Let's check the horizontal shift. For \( y=\lfloor x + 2
floor \), when \( x=-5 \), \( \lfloor -5 + 2
floor=\lfloor -3
floor=-3 \), which matches the top - left point (black dot at \( x = - 5,y=-3 \)). When \( x=-4 \), \( \lfloor -4 + 2
floor=\lfloor -2
floor=-2 \), which matches the black dot at \( x=-4,y = - 2 \). Let's check other options:

• For \( y=\lfloor x

floor \), at \( x=-5 \), \( \lfloor -5
floor=-5
eq - 3 \), so this is wrong.

• For \( y=\lfloor x - 2

floor \), at \( x=-5 \), \( \lfloor -5-2
floor=\lfloor -7
floor=-7
eq - 3 \), so this is wrong.

• For \( y=\lfloor|x|

floor + 2 \), at \( x=-5 \), \( \lfloor|-5|
floor+2=\lfloor5
floor + 2=5 + 2 = 7
eq - 3 \), so this is wrong.

Step2: Confirm the function

By checking the key points on the graph and substituting into the function \( y=\lfloor x + 2
floor \), we find that the function that matches the graph is \( \lfloor x + 2
floor \).

Answer:

\(\lfloor x + 2
floor\) (the option with \(\lfloor x + 2
floor\))