Graph each function. See Examples 1 and 2. ƒ(x) = ⅔ |x|

Verified step by step guidance

1. Recognize that the function ƒ(x) = \(\frac{2}{3}\) |x| involves the absolute value of x, which means the graph will be V-shaped, symmetric about the y-axis.
2. Recall that the absolute value function |x| outputs x when x ≥ 0 and -x when x < 0, so the function can be written as a piecewise function: \[ ƒ(x) = \begin{cases} \frac{2}{3}x & \text{if } x \geq 0 \\ \frac{2}{3}(-x) & \text{if } x < 0 \end{cases} \]
3. Plot key points by choosing values of x, such as -3, -1, 0, 1, and 3, and calculate the corresponding y-values using the function ƒ(x) = \(\frac{2}{3}\) |x|.
4. Draw the graph by connecting the points with two straight lines: one line with slope \(\frac{2}{3}\) for x ≥ 0, and the other line with slope -\(\frac{2}{3}\) for x < 0, meeting at the origin (0,0).
5. Label the graph clearly, noting the vertex at the origin and the slopes of the lines, to complete the graph of ƒ(x) = \(\frac{2}{3}\) |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 and Scaling

Multiplying a function by a constant, such as ⅔ in ƒ(x) = ⅔|x|, vertically scales the graph. A factor less than 1 compresses the graph towards the x-axis, reducing the slope of the lines forming the V-shape. Recognizing this helps in accurately sketching the transformed graph.

Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise linear functions, splitting the domain into x ≥ 0 and x < 0. Graphing involves plotting each linear piece separately and combining them. This approach clarifies the behavior of the function on different intervals.