1.6 Absolute Value Functions

Absolute Value Functions

Absolute value functions produce V-shaped graphs that are always symmetric around a central point called the vertex. They measure magnitude without regard to sign, which makes them useful for modeling distances, tolerances, and any situation where direction doesn't matter.

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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 specific transformation:

• $a$ controls the vertical stretch/compression and direction of opening
  • If $|a| > 1$, the graph is steeper than the parent function
  • If $0 < |a| < 1$, the graph is flatter than the parent function
  • If $a < 0$, the graph opens downward (inverted V-shape)
• $h$ controls the horizontal shift of the vertex
  • Positive $h$ shifts the graph right; negative $h$ shifts it left
  • Watch the sign carefully: $f(x) = |x - 3|$ shifts right 3, because the form is $x - h$
• $k$ controls the vertical shift of the vertex
  • Positive $k$ shifts up; negative $k$ shifts down

The vertex sits at $(h, k)$ and is the point where the graph changes direction. The graph is symmetric about the vertical line $x = h$, called the axis of symmetry.

Domain and Range:

• The domain is all real numbers for every absolute value function.
• The range depends on the sign of $a$:
  • If $a > 0$ (opens up): $y \geq k$
  • If $a < 0$ (opens down): $y \leq k$

Transformations and Parent Function

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

Every absolute value function in the form $f(x) = a|x - h| + k$ is a transformation of this parent. You can also express any absolute value function as a piecewise function. For example:

$f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

This piecewise definition is what creates the two linear "arms" of the V-shape. The absolute value function is continuous for all real numbers, with no breaks or gaps.

Reflections over the y-axis can also apply. For instance, $f(x) = |{-x}| = |x|$, so a y-axis reflection doesn't change the graph of the basic absolute value function. However, for more complex expressions inside the absolute value, horizontal reflections can matter.

Solving Absolute Value Equations

The core idea: if $|A| = B$, then the expression $A$ is either $B$ or $-B$, because both values are the same distance from zero.

Steps to solve:

1. Isolate the absolute value expression on one side of the equation.
2. Check the other side. If it equals a negative number, stop. There's no solution, since absolute value can never be negative.
3. Split into two equations: one where the inside equals the positive value, one where it equals the negative value.
4. Solve each equation separately.
5. Check your solutions in the original equation (especially important when there are additional terms or nested absolute values).

Example: Solve $|x - 3| = 5$

1. The absolute value is already isolated.

2. The right side is 5 (positive), so solutions exist.

3. Split: $x - 3 = 5$ or $x - 3 = -5$

4. Solve: $x = 8$ or $x = -2$

5. Solution set: $\{8, -2\}$

Example with no solution: $|x + 1| = -4$

The right side is negative. Absolute value output is always $\geq 0$, so there is no solution.

When extra steps are needed: If the absolute value isn't isolated, handle that first. For $3|2x - 1| + 7 = 22$, subtract 7 to get $3|2x - 1| = 15$, then divide by 3 to get $|2x - 1| = 5$, and now split into two equations.

Equations with absolute values on both sides, like $|2x - 1| = |x + 2|$, require you to consider the cases where the insides are equal and where they are opposites: $2x - 1 = x + 2$ or $2x - 1 = -(x + 2)$.

Real-World Applications of Absolute Value

Distance on a number line: The distance between any two points $a$ and $b$ is $|a - b|$. For example, the distance between 3 and $-5$ is $|3 - (-5)| = |8| = 8$. This works regardless of which point you subtract from which, since $|a - b| = |b - a|$.

Tolerances and error: Manufacturing and engineering use absolute value to express acceptable deviation. If a part must be 10 cm long with a tolerance of 0.2 cm, the constraint is $|x - 10| \leq 0.2$, meaning the actual length $x$ must fall between 9.8 cm and 10.2 cm.

Magnitude without direction: Absolute value captures "how much" without "which way." A company that lost $50,000 and one that gained $50,000 both have a magnitude of $|\pm 50{,}000| = 50{,}000$. This is useful whenever you care about size rather than sign.