What happens to a graph when a constant is added or subtracted from the x portion of a function?

When a constant is added or subtracted from the x portion of a function, the graph of that function shifts horizontally. This horizontal shift occurs because modifying the x-value of the function effectively changes where the function's output is calculated for a given input.

For example, if you take a function f(x) and consider f(x - c) for some constant c, you're shifting the entire graph to the right by c units if c is positive, and to the left by c units if c is negative. This means that for every input x, the new input is x - c, which causes the output to shift along the x-axis rather than changing the height of the graph, as would be the case if the constant were added or subtracted from the entire function instead (which would result in a vertical shift).

This concept is essential in understanding the translation properties of functions in the coordinate plane and plays a significant role in function transformations in mathematics.