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

Verified step by step guidance

1. Step 1: Recognize that the function \( f(x) = 3|x| \) is a transformation of the basic absolute value function \( g(x) = |x| \).
2. Step 2: Understand that the coefficient 3 in front of \(|x|\) indicates a vertical stretch of the graph by a factor of 3. This means that for every point \((x, y)\) on the graph of \(g(x) = |x|\), the corresponding point on \(f(x) = 3|x|\) will be \((x, 3y)\).
3. Step 3: Identify the vertex of the graph. Since there are no horizontal or vertical shifts, the vertex remains at the origin \((0, 0)\).
4. Step 4: Plot key points. For example, calculate \(f(x)\) for \(x = -1, 0, 1\) to get points \((-1, 3), (0, 0), (1, 3)\).
5. Step 5: Draw the graph by connecting the points with straight lines, forming a 'V' shape that opens upwards, with the vertex at the origin and the arms of the 'V' being steeper than those of \(g(x) = |x|\) due to the vertical stretch.

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, reflecting the symmetry about the y-axis.

Vertical Stretch

A vertical stretch occurs when a function is multiplied by a constant factor greater than one. In the function ƒ(x) = 3|x|, the factor of 3 stretches the graph vertically by a factor of 3, making it steeper than the basic absolute value function. This transformation affects the y-values of the function, increasing them proportionally while maintaining the overall shape of the graph.

Graphing Techniques

Graphing techniques involve plotting points and understanding transformations to visualize functions accurately. For ƒ(x) = 3|x|, one can start by plotting key points such as (0,0), (1,3), and (-1,3), then connect these points to form the V-shape. Familiarity with transformations, such as shifts and stretches, is essential for accurately representing the function on a coordinate plane.