The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.
(a) y = ƒ(x) +3

Key Concepts

Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. In this case, adding a constant to the function, such as in y = ƒ(x) + 3, results in a vertical shift of the graph. Specifically, the entire graph of ƒ(x) is moved upward by 3 units, affecting all y-values while keeping the x-values unchanged.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visually represent the relationship between the input (x-values) and output (y-values) of a function. Understanding how to read and interpret graphs is crucial, as it allows one to identify key features such as intercepts, maxima, minima, and the overall shape of the function, which are essential for accurately sketching transformed functions.

Vertical Shift

A vertical shift occurs when a function's graph is moved up or down on the coordinate plane. This shift is determined by the constant added or subtracted from the function's output. For example, in the function y = ƒ(x) + 3, every point on the graph of ƒ(x) is raised by 3 units, which alters the y-coordinates of all points while leaving the x-coordinates unchanged, effectively translating the graph vertically.