Graph each function. See Examples 1 and 2. ƒ(x)=2/3|x|

Verified step by step guidance

1. Recognize that the function is a transformation of the absolute value function, given by \(f(x) = \frac{2}{3} |x|\). The absolute value function \(|x|\) creates a V-shaped graph symmetric about the y-axis.
2. Identify the key points of the parent function \(y = |x|\), such as \((0,0)\), \((1,1)\), and \((-1,1)\). These points will help in plotting the transformed graph.
3. Apply the vertical stretch/compression factor \(\frac{2}{3}\) to the y-values of the key points. For example, the point \((1,1)\) on \(y=|x|\) becomes \((1, \frac{2}{3} \times 1) = (1, \frac{2}{3})\) on \(f(x)\).
4. Plot the transformed points on the coordinate plane: \((0,0)\) remains the same, \((1, \frac{2}{3})\), and \((-1, \frac{2}{3})\). Connect these points with straight lines forming a V shape.
5. Label the graph clearly, noting that the vertex is at the origin and the graph opens upwards with a slope of \(\frac{2}{3}\) on both sides, reflecting the vertical compression compared to \(y=|x|\).

Key Concepts

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x regardless of its sign. It creates a V-shaped graph symmetric about the y-axis, with the vertex at the origin (0,0). Understanding this shape is essential for graphing functions involving absolute values.

Function Transformation - Vertical Scaling

Vertical scaling involves multiplying the output of a function by a constant factor. For ƒ(x) = (2/3)|x|, the graph of |x| is stretched or compressed vertically by 2/3, making it less steep than the basic absolute value graph. This changes the slope of the lines forming the V-shape.

Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise functions, splitting the domain into parts where the function behaves differently. For |x|, it equals x when x ≥ 0 and -x when x < 0. Recognizing this helps in plotting the graph accurately by considering each piece separately.