3.6 Absolute Value Functions

Graphing and Solving Absolute Value Functions

Absolute value functions measure distance from zero, regardless of direction. They produce V-shaped graphs that can be shifted, stretched, or flipped using transformations. Once you understand how each part of the equation controls the graph, you can sketch these quickly and accurately.

Solving absolute value equations means considering two cases: the expression inside could be positive or negative. This section also covers real-world uses like measuring distances and setting manufacturing tolerances.

Key Features of Absolute Value Graphs

The general form of an absolute value function is:

$f(x) = a|x - h| + k$

Each parameter controls a different transformation:

• $a$ controls the vertical stretch/compression and orientation
  • $a > 1$: graph is steeper (vertically stretched)
  • $0 < a < 1$: graph is flatter (vertically compressed)
  • $a < 0$: graph flips upside down, opening downward
• $h$ controls the horizontal shift of the vertex
  • $h > 0$: shifts right
  • $h < 0$: shifts left
  • Watch the sign carefully: $f(x) = |x - 3|$ shifts right 3 units, because the form is $x - h$
• $k$ controls the vertical shift of the vertex
  • $k > 0$: shifts up
  • $k < 0$: shifts down

The vertex is the point $(h, k)$ where the graph changes direction. When $a > 0$, the vertex is the minimum point (bottom of the V). When $a < 0$, the vertex is the maximum point (top of an upside-down V).

The graph is always symmetric about the vertical line $x = h$. This axis of symmetry passes through the vertex.

Parent Function and Transformations

The parent function is $f(x) = |x|$, which has its vertex at $(0, 0)$ and opens upward with a slope of 1 on the right side and $-1$ on the left side.

From there, transformations build on each other:

• Vertical stretch/compression: $a|x|$ (e.g., $2|x|$ is twice as steep)
• Reflection: $-|x|$ flips the V upside down
• Horizontal shift: $|x - h|$ moves the vertex left or right
• Vertical shift: $|x| + k$ moves the vertex up or down

For example, $f(x) = -2|x + 1| + 5$ means: start with $|x|$, stretch vertically by 2, flip it upside down, shift left 1 unit, and shift up 5 units. The vertex is at $(-1, 5)$.

The domain of any absolute value function is all real numbers. The range depends on the vertex and orientation:

• If $a > 0$: range is $[k, \infty)$
• If $a < 0$: range is $(-\infty, k]$

Solving Absolute Value Equations

To solve an equation like $|2x - 6| = 10$:

1. Isolate the absolute value on one side of the equation. If there are terms outside the absolute value, move them first.

2. Check the other side. If the absolute value equals a negative number, there is no solution (absolute value can never be negative).

3. Set up two cases:

   • Case 1: The expression inside equals the positive value. $2x - 6 = 10$ → $2x = 16$ → $x = 8$
   • Case 2: The expression inside equals the negative value. $2x - 6 = -10$ → $2x = -4$ → $x = -2$

4. Check both solutions by substituting back into the original equation.

   • $|2(8) - 6| = |10| = 10$ ✓
   • $|2(-2) - 6| = |-10| = 10$ ✓

Both solutions work, so $x = 8$ and $x = -2$.

A common mistake is forgetting to isolate the absolute value first. If you have $3|x - 1| + 4 = 19$, subtract 4 and divide by 3 before splitting into two cases.

Real-World Applications of Absolute Value

Distance is naturally an absolute value concept. Distance is always positive regardless of direction. Walking 3 miles east and then 2 miles west gives a total distance of $|3| + |-2| = 5$ miles.

Tolerance in manufacturing uses absolute value to express acceptable deviation from a target. If a machine part should be 10 cm long with a tolerance of $\pm 0.2$ cm, the acceptable lengths satisfy:

$|x - 10| \leq 0.2$

This means $x$ must fall between 9.8 cm and 10.2 cm. To solve it, split into $-0.2 \leq x - 10 \leq 0.2$, then add 10 to all parts.