Slope
Definition
The slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it has been widely used ever since.
Mathematical Description
The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run. 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.
In mathematical terms, the slope m of the line is calculated as:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are coordinates of two distinct points on the line.
Types of Slope
There are four types of slopes:
- Positive Slope: If the line moves upward from left to right, then the line has a positive slope.
- Negative Slope: If the line moves downwards from left to right, then the line has a negative slope.
- Zero Slope: If a line is horizontal the slope is zero. This is a constant function.
- Undefined Slope: If a line is vertical the slope is undefined (the "rise" is infinite).
Slope in Different Fields
The concept of slope is used in various fields of science and engineering, and the interpretation of the slope depends on the context.
Physics
In physics, the slope of the line on a position-time graph as it relates to velocity. The slope of the line on a velocity-time graph as it relates to acceleration.
Economics
In economics, the slope of a line on a supply and demand graph as it relates to the elasticity of a good or service.
Geography
In geography, the slope of a line on a topographic map as it relates to the steepness of a hill.
Calculating Slope
The slope of a line can be calculated by using the formula:
m = (y2 - y1) / (x2 - x1)
This formula is known as the slope formula.
Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a line. It is written in the form:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Point-Slope Form
The point-slope form of a linear equation is another way to express the equation of a line. It is derived from the slope formula and is written as:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is a point on the line.