The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.
(c) y = ƒ(x+3) - 2

Key Concepts

Function Transformations

Function transformations involve altering the graph of a function through shifts, stretches, or reflections. In this case, the transformation y = ƒ(x + 3) - 2 indicates a horizontal shift to the left by 3 units and a vertical shift downward by 2 units. Understanding these transformations is crucial for accurately sketching the new graph based on the original function.

Horizontal Shifts

A horizontal shift occurs when the input of a function is adjusted by adding or subtracting a constant. For the function y = ƒ(x + 3), the graph shifts left by 3 units. This means that every point on the original graph will move leftward, affecting the x-coordinates of all key points while keeping their y-coordinates unchanged.

Vertical Shifts

A vertical shift modifies the output of a function by adding or subtracting a constant. In the transformation y = ƒ(x + 3) - 2, the graph shifts downward by 2 units. This adjustment affects the y-coordinates of all points on the graph, resulting in a new position for each point while maintaining the same horizontal alignment.