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

Verified step by step guidance

1. Step 1: Understand the function ƒ(x) = \frac{2}{3} |x|. This is a piecewise function that represents a V-shaped graph, where the vertex is at the origin (0,0).
2. Step 2: Recognize that the function is a transformation of the basic absolute value function |x|. The coefficient \frac{2}{3} affects the steepness of the V-shape.
3. Step 3: Identify the effect of the coefficient \frac{2}{3}. It compresses the graph vertically, making it less steep compared to the standard |x| graph.
4. Step 4: Plot key points. Start with the vertex at (0,0). Then choose points on either side of the vertex, such as (1, \frac{2}{3}) and (-1, \frac{2}{3}), to see the effect of the transformation.
5. Step 5: Draw the graph by connecting the points with straight lines, forming a V-shape. The lines should be less steep than the lines of the graph of |x| due to the \frac{2}{3} factor.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This means that for any real number x, |x| is equal to x if x is positive or zero, and -x if x is negative. Understanding this function is crucial for graphing, as it creates a V-shaped graph that opens upwards.

Vertical Scaling

Vertical scaling involves multiplying a function by a constant factor, which affects the height of the graph. In the function ƒ(x) = ⅔ |x|, the factor ⅔ compresses the graph vertically, making it less steep than the standard absolute value graph. This concept is essential for accurately representing how the function behaves compared to its parent function.

Graphing Techniques

Graphing techniques include plotting key points, understanding symmetry, and recognizing transformations. For ƒ(x) = ⅔ |x|, one would start by plotting points such as (0,0), (1,⅔), and (-1,⅔), and then use the symmetry of the absolute value function to complete the graph. Mastery of these techniques is vital for accurately visualizing the function's behavior.