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|

Verified step by step guidance

1. Start by graphing the parent function f(x) = |x|. This is a V-shaped graph with its vertex at the origin (0, 0). The graph opens upwards, and the slope of the lines on either side of the vertex is 1 and -1, respectively.
2. Next, analyze the transformation applied to the function h(x) = -|x+3|. The term (x+3) inside the absolute value indicates a horizontal shift. Specifically, the graph of f(x) = |x| is shifted 3 units to the left. This means the vertex of the graph moves from (0, 0) to (-3, 0).
3. The negative sign outside the absolute value, -|x+3|, reflects the graph across the x-axis. This means the V-shaped graph now opens downward instead of upward.
4. Combine the transformations: Start with the graph of f(x) = |x|, shift it 3 units to the left, and then reflect it across the x-axis. The vertex of the transformed graph is at (-3, 0), and the slopes of the lines on either side of the vertex are -1 and 1, respectively.
5. Plot the final graph of h(x) = -|x+3| using the transformations described. Ensure the vertex is at (-3, 0), the graph opens downward, and the slopes of the lines are consistent with the reflection and shift.

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.

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. Recognizing these transformations allows for accurate graphing of modified functions.

Graphing Techniques

Graphing techniques include plotting key points, identifying transformations, and understanding the behavior of functions. For h(x) = -|x+3|, one must first graph f(x) = |x|, apply the transformations, and then accurately depict the new graph. Mastery of these techniques is essential for visualizing and interpreting functions.