# Absolute Value Function — Graph, Properties, Examples

The absolute value function f(x)=∣x∣ outputs the distance of x from zero — always non-negative. Its graph is a V centered at the origin. This article covers the piecewise definition, the V-shape, transformations, three worked examples, and the most common slip in solving absolute-value equations.

## A Function That Forgets the Sign

The absolute value of a number is its distance from zero — and distances are never negative. The absolute value function applies this to every x in the domain, producing a graph that looks like the letter V.

The V-shape is what makes the absolute value function distinctive — it has a _kink_ (a sharp corner) at the vertex. The kink is what makes the function continuous everywhere but _non-differentiable_ at the vertex. Calculus students meet this fact early and revisit it often.

## What the Absolute Value Function Is

The **absolute value function** is defined piecewise:

f(x)=∣x∣={x if x≥0,−x if x<0.

In plain language: when x is non-negative, the function returns x unchanged. When x is negative, the function returns −x (which is positive). The output is never negative.

The domain is all real numbers R; the range is [0,∞).

> **Quick facts.**
> 
> - **Definition:** ∣x∣=x if x≥0; ∣x∣=−x if x<0.
> - **Domain:** R. **Range:** [0,∞).
> - **Shape:** V, with vertex at (0,0) for the parent function.
> - **Vertex form:** f(x)=a∣x−h∣+k has vertex (h,k).
> - **Symmetry:** about the y-axis (the function is even).
> - **Continuity / differentiability:** continuous everywhere; not differentiable at the vertex.

## Properties of the Absolute Value Function

**1. Always non-negative.** ∣x∣≥0 for every real x. The only zero is at x=0.

**2. Even function.** ∣−x∣=∣x∣. The graph is symmetric about the y-axis.

**3. Triangle inequality.** ∣a+b∣≤∣a∣+∣b∣. Adding two numbers can only shorten the resulting distance compared to the sum of individual distances.

**4. Multiplicative.** ∣ab∣=∣a∣⋅∣b∣. The absolute value distributes over multiplication.

**5. Reciprocal of non-zero.** ∣1/a∣=1/∣a∣ for a≠0.

**6. Equation with absolute value.** ∣x∣=c has solutions x=c or x=−c when c≥0; no solution when c<0.

## Transformations: f(x)=a∣x−h∣+k

The general absolute-value function carries four transformation parameters.

- **a** — vertical stretch (or compression). |a|>1 makes the V steeper; |a|<1 makes it shallower. a<0 flips the V upside down.

- **h** — horizontal shift. The vertex moves to x=h.

- **k** — vertical shift. The vertex moves to y=k.

The vertex is always at (h,k). The V opens upward when a>0, downward when a<0.

**Example.** f(x)=−2∣x−1∣+4: vertex at (1,4), opens downward, twice as steep as the parent V.

## Examples of Absolute Value Function

**Quick.** Evaluate ∣−7∣ and ∣5−9∣.

∣−7∣=7. ∣5−9∣=∣−4∣=4.

**Final answer:** ∣−7∣=7 and ∣5−9∣=4.

**Standard (Wrong Path First — The Mistake Worth Making Once).** Solve ∣2x−3∣=7.

_The wrong path._ The rusher writes 2x−3=7 and solves to get x=5. They submit "x=5" as the answer.

The flaw: the inside of the absolute value, 2x−3, could equal 7 _or_ −7 — both give the same absolute value.

_The rescue._ Split into two cases.

Case 1: 2x−3=7⟹2x=10⟹x=5.

Case 2: 2x−3=−7⟹2x=−4⟹x=−2.

Check both: ∣2(5)−3∣=∣7∣=7 and ∣2(−2)−3∣=∣−7∣=7.

**Final answer:** x=5 or x=−2.

**Stretch.** Find the vertex and graph f(x)=−3∣x+2∣+5.

Vertex: (−2,5). Opens downward (since a=−3<0). Slope of the arms: ±3.

**Final answer:** vertex (−2,5), opening downward, arms with slopes ±3.

## Where the Absolute Value Function Shows Up

The absolute value function captures the idea of _distance_ — and distance is one of the most useful concepts in applied mathematics.

- **Error and tolerance.** In engineering, ∣x−xtarget∣<ε defines a tolerance zone.
- **Signal processing.** The _magnitude_ of a signal at each moment is its absolute value.
- **Statistics.** Mean absolute deviation (MAD) is the average of ∣xi−xˉ∣.
- **Programming.** The `abs()` function exists in every programming language because distance computations come up everywhere.
- **Calculus.** The absolute value function is the textbook example of a continuous function that is not everywhere differentiable.

## Absolute Value Function: Mistakes Students Make Most Often

### **1. Forgetting the negative case in equations.**

**Where it slips in:** Solving ∣x−3∣=5 — student writes only x−3=5.

**Don't do this:** Drop the negative-inside case.

**The correct way:** ∣x−3∣=5⟹x−3=5 or x−3=−5.

### **2. Distributing absolute value over a sum.**

**Where it slips in:** Writing ∣a+b∣=∣a∣+∣b∣ in general.

**Don't do this:** Apply the absolute value to each term separately.

**The correct way:** ∣a+b∣≤∣a∣+∣b∣ — the triangle inequality. Equality only holds when a and b have the same sign.

### **3. Forgetting that ∣x∣=−5 has no solution.**

**Where it slips in:** Student tries to solve ∣x−2∣=−3.

**Don't do this:** Apply the two-case method to a negative right side.

**The correct way:** Absolute value is never negative. Recognise and stop.

### **4. Misplacing the vertex of a transformed graph.**

**Where it slips in:** f(x)=∣x+3∣ — student places the vertex at (3,0).

**Don't do this:** Read the sign inside the absolute value directly.

**The correct way:** The sign inside is the _opposite_ of the x-coordinate of the vertex.

## The Mathematicians Who Shaped Distance and Magnitude

**Augustin-Louis Cauchy (1789–1857, France)** introduced rigorous definitions of continuity and the absolute value in his analysis lectures (1821).

**Karl Weierstrass (1815–1897, Germany)** used ∣x∣ in his epsilon-delta proofs of continuity.

**Hermann Minkowski (1864–1909, Germany)** generalised the absolute value to higher dimensions, producing the _Minkowski distance_.

## Conclusion

- The **absolute value function** f(x)=∣x∣ returns the distance from zero — always non-negative.
- Its graph is a V with vertex at the origin (for the parent function).
- The transformed form a∣x−h∣+k has vertex at (h,k).
- The single most common mistake is forgetting the negative-inside case when solving absolute-value equations.
- The triangle inequality ∣a+b∣≤∣a∣+∣b∣ blocks naive distribution.

## Quick Self-Check — Try These

1. Evaluate ∣3−8∣ and ∣−12∣.
2. Solve ∣x+4∣=9 (find all values of x).
3. Find the vertex of f(x)=−∣x−6∣+3 and describe the graph.