Lesson 3.6: Absolute Value Functions

New Concept

The absolute value function, $f(x)=|x|$, represents distance from zero. We'll explore how to graph these V-shaped functions with transformations and how to algebraically solve equations like $|A|=B$ by considering both positive and negative cases.

What’s next

You're all set to begin! Next, you'll tackle interactive examples on graphing absolute value functions, followed by practice problems for solving equations.

Property

The absolute value function can be defined as a piecewise function

Examples

• A component is rated at 250 ohms with a tolerance of $\pm5\%$. The range of actual resistance $R$ can be expressed as $|R - 250| \leq 12.5$.

Property

The graph of a transformed absolute value function is given by $f(x) = a|x - h| + k$. The graph is a 'V' shape with its corner point, or vertex, at $(h, k)$. The value of $a$ determines the vertical stretch and direction. If $a > 0$, the graph opens upwards. If $a < 0$, it opens downwards, reflecting across the x-axis. The magnitude of $a$ determines the steepness.

Examples

• An absolute value function is shifted right 5 units and down 2 units. The equation is $f(x) = |x - 5| - 2$, and its vertex is at $(5, -2)$.

• A function is vertically stretched by a factor of 2, shifted left 1 unit, and up 3 units. The equation is $g(x) = 2|x + 1| + 3$.

Property

For real numbers $A$ and $B$, an equation of the form $|A| = B$, with $B \geq 0$, will have solutions when $A = B$ or $A = -B$. If $B < 0$, the equation $|A| = B$ has no solution.

To solve:

1. Isolate the absolute value term.
2. Set the expression inside the absolute value equal to both $B$ and $-B$.
3. Solve both resulting equations for the variable.

Examples

• To solve $|x - 4| = 9$, set up two cases: $x - 4 = 9$ gives $x = 13$, and $x - 4 = -9$ gives $x = -5$.