For AI agents: use /llms.txt for the Nakafa content index.
An absolute value function is a function that produces positive or zero values from any input, regardless of the original sign of the input. Geometrically, absolute value can be understood as the distance of a number from the zero point on the number line.
For any real number , the absolute value function is defined as:
Components of absolute value functions:
Absolute value functions have several important properties that need to be understood:
Basic properties:
Triangle inequality properties:
The following is a visualization of the basic absolute value function:
Value table for function :
Absolute value functions can be transformed in various ways:
The function shifts the graph upward (if ) or downward (if ).
The function shifts the graph to the right (if ) or to the left (if ).
The function changes the slope of the graph:
To make it easier to understand, let's look at the following example:
The general form of an absolute value function is:
where:
Transformation table:
| Parameter | Value | Effect on Graph |
|---|---|---|
| Positive, | Graph becomes steeper | |
| Positive, | Graph becomes gentler | |
| Negative | Graph is inverted | |
| Positive | Shift to the right | |
| Negative | Shift to the left | |
| Positive | Shift upward | |
| Negative | Shift downward |
Solving absolute value equations:
To solve with :
Example: Solve
Solving absolute value inequalities:
For with :
For with :
Determine the value of for
Solve the equation
Solve the inequality
Determine the vertex of the function
The distance between two cities is . If city is located at coordinate , where is city located?
Calculating function values for various inputs:
Substitute each value of into the function :
Solving absolute value equations:
For the equation , we use the definition of absolute value which produces two possibilities:
Solving absolute value inequalities:
For , we use the property that is equivalent to :
Determining the vertex:
From the function , we can identify the parameters:
Calculating position based on distance:
Given that the distance between cities and is , with city at coordinate . Let city be at coordinate :
So the solution set is .
The vertex is located at .
So city can be located at coordinate or .
