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

Verified step by step guidance

1. Recognize that the function is given by \(g(x) = |x| - 1\), which involves the absolute value function \(|x|\). The absolute value function outputs the distance of \(x\) from zero, so it is always non-negative.
2. Recall the basic shape of the graph of \(y = |x|\): it forms a 'V' shape with its vertex at the origin \((0,0)\), where the graph is symmetric about the y-axis.
3. To graph \(g(x) = |x| - 1\), start by taking the graph of \(y = |x|\) and then shift it downward by 1 unit because of the '-1' outside the absolute value.
4. Plot key points to help sketch the graph: for example, at \(x=0\), \(g(0) = |0| - 1 = -1\); at \(x=1\), \(g(1) = |1| - 1 = 0\); and at \(x=-1\), \(g(-1) = |-1| - 1 = 0\). These points help define the vertex and the arms of the 'V'.
5. Draw the graph by connecting these points with two straight lines forming a 'V' shape, with the vertex at \((0, -1)\) and the arms extending upward with slope 1 to the right and slope -1 to the left.

Key Concepts

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x, making all inputs positive or zero. Its graph forms a 'V' shape with the vertex at the origin, reflecting negative inputs across the y-axis.

Function Transformation

Function transformations involve shifting, stretching, or reflecting graphs. For g(x) = |x| - 1, subtracting 1 shifts the entire graph of |x| downward by one unit, moving the vertex from (0,0) to (0,-1).

Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise functions, defining different expressions for x ≥ 0 and x < 0. Understanding this helps in plotting points accurately and visualizing the graph's shape.