functions

Absolute Value Functions

Algebra I25 minDifficulty

You should know: absolute value equations, function transformations, domain and range

Overview

The absolute value function f(x) = |x| produces a V-shaped graph with vertex at the origin. The general form f(x) = a|x - h| + k shifts the vertex to (h, k) and scales the arms. Like parabolas, |a| > 1 narrows the V and |a| < 1 widens it; a < 0 reflects it downward. The function is piecewise linear.

Intuition

f(x) = |x| reflects any negative input to positive: left of 0, the graph goes up with slope -1; right of 0, slope +1. The result is a V-shape. f(x) = |x - 3| + 2 moves the V right by 3 and up by 2, vertex at (3, 2). Negative a flips the V to an inverted V (like a mountain instead of a valley).

Formal Definition

Definition

f(x) = a|x - h| + k, vertex (h, k). As a piecewise function: f(x) = a(x-h) + k if x >= h, a(-(x-h)) + k = -a(x-h)+k if x < h. Domain: all reals. Range: if a > 0, range = [k, inf); if a < 0, range = (-inf, k]. The vertex is the minimum (a > 0) or maximum (a < 0).

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

Vertex form of absolute value function

|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}

Piecewise definition

Notation

Notation	Meaning
$\|x\|$	Absolute value of x
$(h, k)$	Vertex of the V-shape

Theorems

Theorem 1: Triangle Inequality

|a + b| <= |a| + |b| for all real a, b. Equality holds when a and b have the same sign (or one is zero).

Worked Examples

1
a = 2, h = -1, k = -3. Vertex at (-1, -3), opens upward (a > 0), steeper than |x|.
$\text{Vertex: } (-1, -3),;\text{slopes: } \pm 2$
2
Range: [-3, inf).
$[-3, \infty)$

✓ Answer

V-shape with vertex (-1, -3), arms with slopes 2 and -2.

Practice Problems

Easyfill in blank

Find the vertex of f(x) = -3|x - 4| + 7.

Common Mistakes

Common Mistake

Treating the absolute value function as a parabola (curved graph).

The graph of |x| is a V-shape made of two straight line segments -- piecewise linear, not curved.

Quiz

The range of f(x) = -|x| + 5 is:

[5, inf)(-inf, 5](-inf, inf)[0, 5]

Historical Background

Absolute value as a concept appears in medieval Arabic mathematics. The modern notation |x| was introduced by Karl Weierstrass in the 19th century. The graph of |x| became pedagogically important as the simplest example of a piecewise-defined function and a function that is continuous but has a non-differentiable corner.

1841
Weierstrass introduces the absolute value notation |x|
Karl Weierstrass

Summary

f(x) = a|x-h| + k: V-shape with vertex (h, k).
a > 0: opens up, minimum k. a < 0: opens down, maximum k.
|a| > 1: steeper. |a| < 1: wider.
Piecewise: f(x) = a(x-h)+k for x >= h, -a(x-h)+k for x < h.

References

BookHall, B. and Fabricant, M. Algebra 1. Prentice Hall, 2001.