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. g(x) = -|x + 4| +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 applying transformations to graph other functions.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, adding a constant inside the absolute value affects horizontal shifts, while adding outside affects vertical shifts. In the function g(x) = -|x + 4| + 2, the transformations include a horizontal shift left by 4 units, a reflection across the x-axis, and a vertical shift up by 2 units.

Graphing Techniques

Graphing techniques involve plotting points and understanding how transformations affect the shape and position of a graph. For the function g(x), one must first graph f(x) = |x|, then apply the identified transformations step-by-step. This systematic approach helps visualize the final graph accurately and understand the relationship between the original and transformed functions.