Absolute value functions are like mathematical boomerangs. They always bounce back, creating a V-shaped graph that's symmetric around a central point. These functions have unique properties that make them useful for modeling real-world situations.

Understanding absolute value functions unlocks a world of practical applications. From calculating distances to analyzing financial data, these functions help us make sense of situations where the magnitude matters more than the direction. They're essential tools in your mathematical toolkit.

Absolute Value Functions

Key features of absolute value graphs

• The general form of an absolute value function is $f(x) = a|x - h| + k$ (also known as the modulus function)
  • $a$ determines the vertical stretch or compression of the graph
    • If $|a| > 1$, the graph is vertically stretched (appears steeper)
    • If $0 < |a| < 1$, the graph is vertically compressed (appears flatter)
    • If $a < 0$, the graph opens downward (inverted V-shape)
  • $h$ represents the horizontal shift of the vertex
    • If $h > 0$, the graph shifts to the right (positive direction)
    • If $h < 0$, the graph shifts to the left (negative direction)
  • $k$ represents the vertical shift of the vertex
    • If $k > 0$, the graph shifts up (positive direction)
    • If $k < 0$, the graph shifts down (negative direction)
• The graph of an absolute value function is V-shaped with a vertex at $(h, k)$
  • The vertex is the point where the graph changes direction
  • The graph is symmetric about the vertical line passing through the vertex
• The domain of an absolute value function is all real numbers
  • The function is defined for any input value ($x$ can be any real number)
• The range of an absolute value function is $y \geq k$ if $a > 0$, or $y \leq k$ if $a < 0$
  • If the graph opens upward, the range is all $y$ values greater than or equal to the vertex's $y$-coordinate
  • If the graph opens downward, the range is all $y$ values less than or equal to the vertex's $y$-coordinate

Transformations and Parent Function

• The parent function of absolute value is $f(x) = |x|$
• Transformations of the parent function include:
  • Vertical and horizontal shifts
  • Vertical and horizontal stretches or compressions
  • Reflections over the x-axis or y-axis
• The absolute value function is continuous for all real numbers

Solving absolute value equations

• To solve an absolute value equation, isolate the absolute value term on one side of the equation
  • If the other side is positive, split the equation into two and solve each part
    • Example: $|x - 3| = 5$ becomes $x - 3 = 5$ or $x - 3 = -5$
      • Solve $x - 3 = 5$ by adding 3 to both sides: $x = 8$
      • Solve $x - 3 = -5$ by adding 3 to both sides: $x = -2$
      • The solution set is $\{8, -2\}$
  • If the other side is negative, there is no solution
    • Example: $|x + 1| = -4$ has no solution because the absolute value is always non-negative
• When solving absolute value equations with variables on both sides, isolate one absolute value term and then follow the above steps
  • Example: $|2x - 1| + 3 = |x + 2|$ becomes $|2x - 1| = |x + 2| - 3$
    • Simplify the right side: $|2x - 1| = |x + 2| - 3$
    • Split into two equations and solve each part
• Absolute value functions can also be expressed using a piecewise definition

Real-world applications of absolute value

• Absolute value can represent the distance between two points on a number line
  • Example: The distance between 3 and -5 is $|3 - (-5)| = |3 + 5| = 8$
    • This concept can be applied to find the distance between any two points (locations, temperatures)
• Absolute value can model situations where the direction is irrelevant, but the magnitude is important
  • Example: A company's profit or loss can be represented using absolute value, as the magnitude is more important than whether it's a profit (positive) or loss (negative)
    • If a company's profit/loss is represented by $x$, then $|x|$ gives the magnitude of the profit/loss
• Absolute value can be used to find the minimum or maximum distance between two functions
  • Example: To find the minimum distance between $f(x) = x^2$ and $g(x) = -x + 2$, set up an absolute value equation: $|x^2 - (-x + 2)| = d$
    • Solve for $x$ when the distance $d$ is minimized (vertex of the absolute value graph)
    • This technique can be used to optimize distances in various applications (network analysis, resource allocation)