Transformations of the Absolute Value Function - Algebra-1

1. Fundamental Concepts

Definition: The absolute value function, denoted as $$f(x) = |x|$$, represents the distance of a number from zero on the number line.
Transformations: Transformations of the absolute value function include shifts, stretches, and reflections which alter the graph's position and shape.
Shifts: Horizontal and vertical shifts are represented by adding or subtracting constants inside or outside the absolute value expression.

Vertical Shift: $$f(x) = |x| + k$$

Adding $k$ shifts the graph up by $k$ units if $k > 0$, and down by $|k|$ units if $k < 0$.

Horizontal Shift: $$f(x) = |x - h|$$

Subtracting $h$ shifts the graph right by $h$ units if $h > 0$, and left by $|h|$ units if $h < 0$.

Reflection: $$f(x) = -|x|$$

Multiplying by $-1$ reflects the graph over the x-axis.

Problem: Graph the function $$f(x) = |x| + 3$$

Validation: Check points such as (0, 3), (-1, 4), and (1, 4).

Problem: Graph the function $$f(x) = |x - 2| - 1$$

Validation: Check points such as (2, -1), (1, 0), and (3, 0).

Graphical Analysis: Use a coordinate plane to visualize transformations step-by-step.
Point Substitution: Substitute specific values into the transformed function to find key points.
Pattern Recognition: Identify patterns in how transformations affect the graph’s shape and position.