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

Verified step by step guidance

1. Understand the function given: \(f(x) = 3|x|\). This means the output is three times the absolute value of \(x\).
2. Recall the shape of the basic absolute value function \(|x|\), which forms a 'V' shape with its vertex at the origin (0,0).
3. To graph \(f(x) = 3|x|\), multiply the output of \(|x|\) by 3, which vertically stretches the graph by a factor of 3. This makes the 'V' shape steeper.
4. Plot key points to guide your graph: for example, when \(x=0\), \(f(0) = 3|0| = 0\); when \(x=1\), \(f(1) = 3|1| = 3\); and when \(x=-1\), \(f(-1) = 3| -1| = 3\).
5. Draw the graph by connecting these points with straight lines forming a 'V' shape, ensuring the vertex is at the origin and the arms rise steeply due to the factor of 3.

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, reflecting all negative inputs as positive outputs.

Vertical Stretching of Functions

Multiplying a function by a constant greater than 1, like 3 in 3|x|, stretches the graph vertically. This means all output values are scaled by that factor, making the graph steeper compared to the parent function.

Graphing Piecewise Functions

Absolute value functions can be viewed as piecewise functions with different expressions for x ≥ 0 and x < 0. Understanding this helps in plotting points accurately and visualizing the graph's shape.