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

Verified step by step guidance

1. Start with the parent function f(x) = |x|. This is the absolute value function, which forms a V-shaped graph with its vertex at the origin (0, 0). The graph is symmetric about the y-axis, with the left side decreasing and the right side increasing.
2. Identify the transformation applied to the function. The given function is h(x) = |x + 3| - 2. The term (x + 3) inside the absolute value indicates a horizontal shift, and the -2 outside the absolute value indicates a vertical shift.
3. Apply the horizontal shift. The term (x + 3) means the graph of f(x) = |x| is shifted 3 units to the left. This moves the vertex of the graph from (0, 0) to (-3, 0).
4. Apply the vertical shift. The -2 outside the absolute value means the graph is shifted 2 units downward. This moves the vertex from (-3, 0) to (-3, -2).
5. Sketch the transformed graph. The new graph of h(x) = |x + 3| - 2 retains the V-shape of the parent function but is now centered at the vertex (-3, -2). The left side of the graph decreases with a slope of -1, and the right side increases with a slope of 1.

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. Mastery of these transformations allows for the manipulation of the base graph to create new functions.

Vertex of a Function

The vertex of a function is the point where the graph changes direction, particularly significant in absolute value functions. For h(x) = |x + 3| - 2, the vertex is at the point (-3, -2). Identifying the vertex is essential for accurately graphing transformed functions and understanding their behavior.