The table provided shows how the modulus function transforms input values ($x$) into output values ($|x|$ or $y$). Crucially, $y$ is always zero or positive.
| Input ($x$) | Output ($|x|$) |
| :—: | :—: |
| $x = 0$ | $|x| = 0$ |
| $x = 0.1$ | $|x| = 0.1$ |
| $x = 0.2$ | $|x| = 0.2$ |
| $x = 0.7$ | $|x| = 0.7$ |
| $x = 0.9$ | $|x| = 0.9$ |
| $x = 1$ | $|x| = 1$ |
| $x = 1.2$ | $|x| = 1.2$ |
| $x = 1.5$ | $|x| = 1.5$ |
| Input ($x$) | Output ($|x|$) |
| :—: | :—: |
| $x = -0.1$ | $|x| = 0.1$ |
| $x = -0.2$ | $|x| = 0.2$ |
| $x = -0.9$ | $|x| = 0.9$ |
| $x = -1$ | $|x| = 1$ |
| $x = -1.2$ | $|x| = 1.2$ |
Notice that pairs of opposite numbers (e.g., $x=1$ and $x=-1$) produce the same output value ($y=1$). This is the characteristic of an even function and results in a graph that is symmetric with respect to the $Y$-axis.
The graph of $y = |x|$ is a distinct V-shape with its vertex (the point where the graph changes direction) at the origin $(0, 0)$.
This V-shape is a visual representation of the piecewise definition of the modulus function:
The function is defined as $y = x$. This is a line starting at the origin and increasing with a slope of $1$.
The function is defined as $y = -x$. This is a line approaching the origin from the left, increasing with a slope of $-(-1) = 1$. The negative sign reflects the negative inputs across the $X$-axis, making all $y$-values positive.
Domain: $(-\infty, \infty)$ (Any real number can be an input for $x$).
Range: $[0, \infty)$ (The output $y$ is always non-negative).
Vertex: The minimum point of the graph is at $(0, 0)$.
This simple V-shaped graph is foundational for understanding how transformations (shifts, stretches, and reflections) affect absolute value functions.