Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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

1. Horizontal shift left

2. Vertical shift down

3. Vertical shift up

4. Horizontal shift right

The correct answer is: Vertical shift up

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.