Answer:
The first choice is the one you want
Explanation:
First thing you need to know about this greatest integer graph is that it is aptly called a step graph. It literally looks like stair steps on your calculator: short horizontal lines that are not connected vertically. Really cool graph.
Second thing you need to know is about transformations of functions. ANY side-to-side movement in ANY function will be in a set of parenthesis (or absolute value symbols, or under a radical sign, or inside the greatest integer brackets, etc.) and ANY up or down movement will be either added or subtracted. Added means you move the function up from its starting position, subtracting means you move the function down from its starting position. Since we have no numbers inside the greatest integer brackets, there is no side-to-side movement. Since there is a "-2" after the brackets, we are moving the whole function down.
If you do not know how to graph these without a calculator and you have no idea what this graph looks like, I recommend going to your calculator to see it. First, call up your "y = " window. Next, hit 2nd-->0 (catalog), then hit the x^2 button (this will take you to the letter I in the catalog). Scroll down til you see "int( " and hit that button. It will take you back to the "y = " window. Enter an x after that set of parenthesis and then close it, then hit " - 2 " and then "graph". Your steps should begin to appear. Notice that the horizontal line between x = 0 and x = 1 is at y = -2. The parent graph has this line between x = 0 and x = 1 on y = 0. The -2 in ours moved the graph down from y = 0 to y = -2
Summing up, the first choice is the one you want as your answer.