Encyclopedia
The
slope or the
gradient is commonly used to describe the measurement of the steepness, incline or grade of a straight line. A higher slope value indicates a steeper incline. In normal
UK usage, the gradient of a slope is defined as the ratio of the "
rise" divided by the "
run" between two points on a line. The term Grade is also used for this definition, which can be mathematically stated as the
tangent of the angle of inclination – the ratio of the altitude change to the horizontal distance between any two points on the grade.
Using
calculus, one can calculate the slope of the tangent to a curve at a point.
The concept of slope, and much of this article, applies directly to grades or
gradients in
geography and
civil engineering. In UK construction work, a slope is often called a fall, and measured as an angle, a gradient or as a ratio such as 1 in 80.
Definition of slope
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:
Given two points and , the change in
x from one to the other is
x2 -
x1, while the change in
y is
y2 -
y1. Substituting both quantities into the above equation obtains the following:
Since the
y-axis is vertical and the
x-axis is horizontal by convention, the above equation is often memorized as "rise over run", where ?
y is the "rise" and ?
x is the "run". Therefore, by convention,
m is equal to the change in
y, the vertical coordinate, divided by the change in
x, the horizontal coordinate; that is,
m is the ratio of the changes. This concept is fundamental to
algebra, analytic geometry,
trigonometry, and
calculus.
Note that the points chosen and the order in which they are used is irrelevant; the same line will always have the same slope. Other curves have "
accelerating" slopes and one can use
calculus to determine such slopes.
Example 1
Suppose a line runs through two points:
P and
Q. By dividing the difference in
y-coordinates by the difference in
x-coordinates, one can obtain the slope of the line:
The slope is
1/2 = 0.5.
Example 2
If a line runs through the points and then:
Geometry
The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. A vertical line does not have a real number slope, since its slope would have to be infinite.
The angle ? a line makes with the positive
x axis is closely related to the slope
m via the
tangent function:
and
.
Two lines are parallel if and only if their slopes are equal or if they both are vertical and therefore undefined; they are
perpendicular if and only if the product of their slopes is -1 or one has a slope of 0 and the other is vertical and undefined.
Slope of a road, etc.
There are two common ways to describe how steep a road is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway. The formula for converting a slope in percentage to degrees is:
Thus, a 100% slope is 45°.
Algebra
If
y is a
linear function of
x, then the coefficient of
x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form
then
m is the slope. This form of a line's equation is called the
slope-intercept form, because
b can be interpreted as the y-intercept of the line, the
y-coordinate where the line intersects the
y-axis.
If the slope
m of a line and a point on the line are both known, then the equation of the line can be found using the
point-slope formula:
For example, consider a line running through the points and . This line has a slope,
m, of
.
One can then write the line's equation, in point-slope form:
or:
.
The slope of a linear equation in the general form:
is given by the formula:
.
Calculus
The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The
derivative of the function at a point is the slope of the line
tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.
Why calculus is necessary
If we let ?
x and ?
y be the distances between two points on a curve, then the slope given by the above definition,,
is the slope of a
secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.
For example, the slope of the secant intersecting
y =
x² at and is
m = / = 3 .
By moving the two points closer together so that ?
y and ?
x decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the
limit, or the value that ?
y/?
x approaches as ?
y and ?
x get closer to zero; it follows that this limit is the exact slope of the tangent. If
y is dependent on
x, then it is sufficient to take the limit where only ?
x approaches zero. Therefore, the slope of the tangent is the limit of ?
y/?
x as ?
x approaches zero. We call this limit the
derivative.
See also
