Final answer:
The transformations f(x) + 3 and g(x) - 3 both affect the dependent variable, resulting in the graph shifting vertically up by 3 units for the first function and down by 3 units for the second function, with no effect on the independent variable.
Step-by-step explanation:
When analyzing the transformations of a function's graph, it is important to understand the roles of the independent variable and the dependent variable. The independent variable is represented by the x-axis, and the dependent variable is represented by the y-axis. In the context of a function expressed as y = f(x), the y-value depends on the x-value.
In the first function, f(x) + 3, the transformation is applied to the dependent variable as it alters the output values of the function. This results in a vertical shift up by 3 units. For the second function, g(x) - 3, similarly, the transformation is on the dependent variable, which results in a vertical shift down by 3 units. It's important to note that no changes have been made to the independent variable (x) in either function. Thus, the transformations do not affect the x-values or the horizontal position of the graph.