### Function Definition The function \( g(x) = 2|x + 1| - 4 \) is a piecewise linear equation involving an absolute value operation. Below is an explanation of each component of the function: - **Absolute Value**: \( |x + 1| \) represents the absolute value of the expression \( x + 1 \). This operation ensures the value inside the absolute value is always non-negative. - **Linear Transformation**: The expression \( 2|x + 1| \) indicates that the absolute value is being scaled by a factor of 2. - **Vertical Shift**: The final expression \( 2|x + 1| - 4 \) reflects a downward shift of the graph by 4 units. ### Graph Description The function creates a "V" shaped graph. Here's a breakdown of its characteristics: - **Vertex**: The vertex of the graph of \( g(x) \) is located at the point (-1, -4). This point is derived from setting the expression inside the absolute value \( x + 1 = 0 \) and finding \( g(x) \). - **Symmetry**: The graph is symmetrical about the vertical line \( x = -1 \). - **Slope**: For \( x > -1 \), the slope of the line is 2, indicating the graph increases as you move right. For \( x < -1 \), the slope is -2, as the graph decreases to the left of \( x = -1 \). This piecewise-linear function with an absolute value translates to a vertex-centered, symmetric graph with straightforward linear sections on either side of the vertex.