Graph each function.
$\text{ƒ}\left(x\right)=\mid x\mid -3$

Verified step by step guidance

1. Understand the base function: The function given is ƒ(x) = |x| - 3. The absolute value function |x| creates a V-shaped graph with its vertex at the origin (0,0).
2. Identify the transformation: The '-3' outside the absolute value shifts the entire graph vertically downward by 3 units. This means the vertex of the graph moves from (0,0) to (0,-3).
3. Plot the vertex: Start by plotting the point (0, -3) on the coordinate plane. This is the lowest point of the graph since the absolute value function is always non-negative.
4. Determine additional points: Choose values of x on both sides of zero (for example, x = 1 and x = -1), calculate ƒ(x) by substituting into the function: ƒ(1) = |1| - 3 and ƒ(-1) = |-1| - 3. Plot these points accordingly.
5. Draw the graph: Connect the points with two straight lines forming a V shape, with the vertex at (0, -3). The left arm of the V will slope upward to the left, and the right arm will slope upward to the right.

Key Concepts

Absolute Value Function

The absolute value function, denoted |x|, outputs the distance of x from zero on the number line, always yielding a non-negative result. 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 - Vertical Shift

A vertical shift moves the graph of a function up or down without changing its shape. In ƒ(x) = |x| - 3, subtracting 3 shifts the entire graph of |x| downward by 3 units, moving the vertex from (0,0) to (0,-3). Recognizing vertical shifts helps in accurately plotting the graph.

Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise functions, defining different expressions for x ≥ 0 and x < 0. For |x|, this means f(x) = x if x ≥ 0, and f(x) = -x if x < 0. Understanding this helps in plotting the graph by considering each piece separately.