Absolute value functions

Definition

Absolute value functions are mathematical expressions that measure the distance of a number from zero on the number line, represented as $|x|$. The graph of an absolute value function typically forms a 'V' shape.

5 Must Know Facts For Your Next Test

1. The standard form of an absolute value function is $f(x) = a|bx + c| + d$.
2. The vertex of the absolute value function $f(x) = a|bx + c| + d$ is at the point $\left( -\frac{c}{b},d \right)$.
3. If $a > 0$, the graph opens upwards; if $a < 0$, it opens downwards.
4. Absolute value functions have domain all real numbers and their range depends on the value of $d$ in $f(x) = a|bx + c| + d$.
5. $f(x) = |x|$ has symmetry with respect to the y-axis, making it an even function.

Review Questions

• What is the vertex of the function $f(x) = 3|2x - 4| + 5$?
• Describe how you would graph $f(x) = -2|x + 1|-3$.
• Explain why absolute value functions are considered piecewise functions.

Related terms

Piecewise Function:

A function composed of multiple sub-functions, each defined over a specific interval.

Vertex:

The highest or lowest point on the graph of an absolute value function, located at $\left( -\frac{c}{b},d \right)$ for $f(x)=a|bx+c|+d$.

Even Function:

A function symmetrical about the y-axis, satisfying $f(-x) = f(x)$ for all x in its domain.