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|

Key Concepts

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This function is V-shaped, with its vertex at the origin (0,0), and it reflects any negative input to positive. Understanding this function is crucial as it serves as the foundation for graphing transformations.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this case, the function h(x) = -|x+3| involves a horizontal shift to the left by 3 units and a vertical reflection across the x-axis. Mastery of these transformations allows for the manipulation of the base graph to achieve the desired function.

Graphing Techniques

Graphing techniques include plotting key points, identifying transformations, and understanding the overall shape of the function. For h(x) = -|x+3|, one would start with the graph of f(x) = |x|, apply the transformations, and then accurately sketch the new graph. Proficiency in these techniques is essential for visualizing and interpreting functions.