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

Verified step by step guidance

1. Step 1: Understand the function. The function ƒ(x)=2/3|x| is an absolute value function, which means it will have a 'V' shape. The 2/3 is a vertical stretch factor, meaning the graph will be taller than the standard absolute value function.
2. Step 2: Identify key points. The vertex of the 'V' is at the origin (0,0). Because of the vertical stretch, when x=3, y=2, and when x=-3, y=2. These are key points that will help you draw the graph.
3. Step 3: Draw the graph. Start by plotting the vertex at the origin. Then plot the points (3,2) and (-3,2). Draw a straight line through these points to form the 'V' shape.
4. Step 4: Check your graph. The graph should be symmetric about the y-axis, because the absolute value function is always symmetric. If it's not, check your points and lines.
5. Step 5: Label your graph. Make sure to label the x and y axes, and write the function equation somewhere on or near the graph for reference.

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 creates a V-shaped graph that opens upwards, with the vertex at the origin (0,0). Understanding how absolute values behave is crucial for graphing functions that include them.

Vertical Scaling

Vertical scaling refers to the stretching or compressing of a graph along the y-axis. In the function ƒ(x) = (2/3)|x|, the coefficient 2/3 indicates that the graph of the absolute value function is vertically compressed by a factor of 2/3. This means that for every value of x, the output will be two-thirds of what it would be in the standard absolute value function.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of a function's graph based on its algebraic expression. For ƒ(x) = (2/3)|x|, one can start by plotting key points, such as (0,0), (1,2/3), and (-1,2/3), and then connect these points to form the V-shape. Familiarity with these techniques is essential for accurately representing functions visually.