Slope

The slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line. The line may be practical (as for a roadway) or in a diagram.

Slope

The slope of a line in the plane is defined as the rise over the run, $m = \frac{\Delta y}{\Delta x}$.

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation:

$\displaystyle m=\frac { \Delta y }{ \Delta x } =\frac { \text{rise} }{ \text{run} }$

The Greek letter delta, $\Delta$, is commonly used in mathematics to mean "difference" or "change". Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the change in $x$ from one to the other is $x_2-x_1$ (run), while the change in $y$ is $y_2-y_1$ (rise).

Intercept

Using the common convention that the horizontal axis represents a variable $x$ and the vertical axis represents a variable $y$, a $y$-intercept is a point where the graph of a function or relation intersects with the $y$-axis of the coordinate system. It also acts as a reference point for slopes and some graphs.

Intercept

Graph with a $y$-intercept at $(0, 1)$.

If the curve in question is given as $y=f(x)$, the $y$-coordinate of the $y$-intercept is found by calculating $f(0)$. Functions which are undefined at $x=0$ have no $y$-intercept.

Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one $y$-intercept. Because functions associate $x$ values to no more than one $y$ value as part of their definition, they can have at most one $y$-intercept.

Analogously, an $x$-intercept is a point where the graph of a function or relation intersects with the $x$-axis. As such, these points satisfy $y=0$. The zeros, or roots, of such a function or relation are the $x$-coordinates of these $x$-intercepts.