Absolute value functions are all about measuring distance from zero, regardless of direction. They create V-shaped graphs that can be shifted, stretched, or flipped. Understanding these transformations helps you predict how the graph will look and behave.

Solving absolute value equations involves considering two cases: when the expression inside is positive or negative. This approach is crucial for finding all possible solutions. Real-world applications include measuring distances and setting tolerances in manufacturing.

Graphing and Solving Absolute Value Functions

Key features of absolute value graphs

• Absolute value function represented by the equation $f(x) = a|x - h| + k$
  • $a$ controls the vertical stretch or compression of the graph
    • $a > 1$ vertically stretches the graph (makes it steeper)
    • $0 < a < 1$ vertically compresses the graph (makes it flatter)
    • $a < 0$ flips the graph upside down, opening downward
  • $h$ represents the horizontal shift of the vertex
    • $h > 0$ shifts the graph to the right ($3$ units right)
    • $h < 0$ shifts the graph to the left ($-2$ units left)
  • $k$ represents the vertical shift of the vertex
    • $k > 0$ shifts the graph up ($2$ units up)
    • $k < 0$ shifts the graph down ($-4$ units down)
• Vertex is the point $(h, k)$ where the graph changes direction
  • $a > 0$ vertex is the minimum point (bottom of the V-shape)
  • $a < 0$ vertex is the maximum point (top of the upside-down V-shape)
• Graph of an absolute value function is symmetric about the vertical line $x = h$

Parent function and transformations

• The parent function for absolute value is f(x) = |x|
• Transformations applied to the parent function:
  • Vertical stretch or compression: a|x|
  • Reflection: -|x|
  • Horizontal shift: |x - h|
  • Vertical shift: |x| + k
• Domain of absolute value functions is all real numbers
• Range depends on the orientation and vertical shift of the function

Solving absolute value equations

• Isolate the absolute value term on one side of the equation
• Consider two cases for the expression inside the absolute value
  • Case 1: Expression is positive or zero
    1. Remove the absolute value symbols
    2. Solve the resulting equation
  • Case 2: Expression is negative
    1. Multiply both sides of the equation by -1
    2. Remove the absolute value symbols
    3. Solve the resulting equation
• Combine solutions from both cases to find all possible solutions
• Check solutions by substituting them back into the original equation

Real-world applications of absolute value

• Distance represents the absolute value between two points on a number line
  • Walking $3$ miles east and $2$ miles west results in a total distance of $|3| + |-2| = 5$ miles
• Tolerance represents the maximum allowed deviation from a target value
  • Machine part length of $10$ cm with a tolerance of $±0.2$ cm has an acceptable range of $|x - 10| ≤ 0.2$, where $x$ is the actual length