Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = 2|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, with a slope of 1 for x > 0 and a slope of -1 for x < 0.
2. Next, analyze the transformation inside the absolute value function. The term (x + 3) 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. Now, consider the coefficient 2 outside the absolute value function. This represents a vertical stretch. The graph of f(x) = |x| is stretched by a factor of 2, meaning the slopes of the lines become steeper: 2 for x > -3 and -2 for x < -3.
4. Combine the transformations: Start with the vertex at (-3, 0), apply the vertical stretch, and ensure the slopes of the graph are adjusted accordingly. The left side of the graph will have a slope of -2, and the right side will have a slope of 2.
5. Finally, sketch the transformed graph h(x) = 2|x + 3| by applying the horizontal shift and vertical stretch to the parent function. Verify that the vertex is at (-3, 0) and the slopes are consistent with the transformations.

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 intersects the origin (0,0) and is symmetric about the y-axis. Understanding this function is crucial as it serves as the foundation for graphing transformations.

Transformations of Functions

Transformations involve shifting, stretching, or reflecting the graph of a function. For the function h(x) = 2|x+3|, the graph of f(x) = |x| is first shifted left by 3 units and then vertically stretched by a factor of 2. 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) = 2|x+3|, one must first graph the base function and then apply the transformations systematically. Mastery of these techniques is essential for visualizing and interpreting the behavior of complex functions.