Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph.

Key Concepts

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. Common transformations include vertical and horizontal shifts, which move the graph up, down, left, or right, and reflections, which flip the graph over an axis. Understanding these transformations is crucial for analyzing how the graph of a function changes based on modifications to its equation.

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, produces a V-shaped graph that opens upwards. It reflects all negative values of x to positive values, resulting in a vertex at the origin (0,0). This function serves as a foundational example for understanding transformations, as various shifts and reflections can be applied to alter its basic shape and position.

Square Root Function

The square root function, represented as g(x) = √x, produces a graph that starts at the origin and increases gradually, forming a curve that extends to the right. This function is essential for understanding transformations, as it can be shifted or reflected to create new graphs. The transformations applied to this function can significantly change its domain and range, affecting how it interacts with other functions.