nonvertical

Nonvertical describes a line in a plane that is not parallel to the vertical axis. In a standard Cartesian coordinate system, this means the line does not run straight up and down; its slope is defined and finite. A nonvertical line can generally be expressed as y = mx + b, where m is the slope and b is the y-intercept.

In implicit form, a line given by ax + by + c = 0 is nonvertical when b ≠ 0, which yields a finite slope m = -a/b. By contrast, a vertical line has the equation x = k, has undefined slope, and does not represent a function y = f(x) of x.

Nonvertical lines include horizontal lines, which have slope m = 0, as well as oblique lines, which have nonzero slopes. The term nonvertical is therefore a way to distinguish lines that can be described as a function of x from those that cannot.

Applications and implications: In graphing and analytic geometry, identifying a line as nonvertical ensures that the line can be graphed as a function of x and that its slope is well-defined. This distinction is also relevant in the vertical line test for functions and in calculus, where differentiability with respect to x applies to nonvertical lines.

In summary, nonvertical refers to lines not parallel to the vertical axis, characterized by a finite slope and expressible in the form y = mx + b or ax + by + c = 0 with b ≠ 0.