What is the effect of adding a constant k to a function f(x)?

When a constant ( k ) is added to a function ( f(x) ), the resulting function becomes ( f(x) + k ). This transformation impacts the vertical position of the function on the graph. Specifically, adding a positive constant ( k ) raises every point on the graph of ( f(x) ) by ( k ) units. As a result, the entire graph shifts upward without altering the shape of the function.

This vertical shift occurs because for each input ( x ), you are now adding ( k ) to the output of ( f(x) ), effectively moving the whole function up by that constant amount. For instance, if you had a linear function that typically runs through the origin and you added 3, every point that was previously at (1, f(1)) would now be at (1, f(1) + 3).

In contrast, other transformations such as a left or right shift involve modifying the input ( x ) directly—leading to a change in the horizontal positioning of the graph rather than a vertical one. Consequently, understanding this principle of adding a constant is crucial for manipulating functions in graphing and function analysis.