In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = |x + 3| - 2

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This function has a V-shaped graph that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial as it serves as the foundation for graphing transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In the case of h(x) = |x + 3| - 2, the graph of f(x) = |x| is shifted left by 3 units and down by 2 units. Recognizing these transformations allows for the accurate graphing of modified functions.

Vertex of a Function

The vertex of a function is the point where the graph changes direction, often representing a minimum or maximum value. For the absolute value function, the vertex is at (0,0). In the transformed function h(x), the vertex shifts to (-3, -2), which is essential for accurately plotting the new graph.