Graph each function. See Examples 6–8.g(x) = |x| - 1

Verified step by step guidance

1. Step 1: Understand the function g(x) = |x| - 1. This is a transformation of the basic absolute value function f(x) = |x|.
2. Step 2: Recognize that the graph of f(x) = |x| is a V-shaped graph with its vertex at the origin (0,0).
3. Step 3: The transformation g(x) = |x| - 1 shifts the graph of f(x) = |x| downward by 1 unit. This means the vertex of the graph will now be at (0, -1).
4. Step 4: Plot the vertex of the graph at (0, -1).
5. Step 5: Draw the V-shaped graph opening upwards from the vertex (0, -1), maintaining the same slope as the original absolute value function.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number x from zero on the number line, always yielding a non-negative result. This function has a V-shaped graph that opens upwards, with its vertex at the origin (0,0). For negative values of x, the function reflects across the y-axis, resulting in positive outputs.

Vertical Shifts

Vertical shifts occur when a function is adjusted up or down on the graph. In the function g(x) = |x| - 1, the '-1' indicates a downward shift of the entire graph of the absolute value function by one unit. This transformation affects the vertex of the graph, moving it from (0,0) to (0,-1).

Graphing Techniques

Graphing techniques involve plotting points and understanding transformations to accurately represent a function visually. For g(x) = |x| - 1, one would start by plotting the basic absolute value function and then apply the vertical shift. Identifying key points, such as the vertex and intercepts, helps in sketching the graph accurately.