Final answer:
The difference in the value of y for each integer change in x in the function y = -9x - 27 is determined by the slope, which is -9. This means that for every increase in x by 1, y decreases by 9, and for every decrease in x by 1, y increases by 9.
Step-by-step explanation:
The question asks about the difference in the value of y for each integer change in x in the function y = -9x - 27. In this linear equation, the coefficient of x, which is -9, represents the slope of the line. The slope indicates the change in y for each one-unit increase in x. Since the slope is -9, for each integer increase in x, the value of y decreases by 9 (because of the negative sign). Conversely, for each integer decrease in x, the value of y increases by 9.
To provide a demonstration, let's choose an arbitrary value for x, let's say x = 2. Plugging this into the equation, y = (-9)(2) - 27 which simplifies to y = -18 - 27, therefore y = -45. Now, if x increases by 1 to x = 3, then y becomes (-9)(3) - 27, which simplifies to y = -27 - 27, hence y = -54. The difference in y values when x increased by 1 is -54 - (-45) = -9, confirming our understanding of the slope.