Slope

In mathematics, the slope of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of said line. With an understanding of algebra and geometry, one can calculate the slope of a straight line; with 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.

Contents

1 Definition of slope 1.1 Example 1 1.2 Example 2 2 Geometry 2.1 Slope of a road, etc. 3 Algebra 4 Calculus 4.1 Why calculus is necessary 5 Related topics

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:

<math>m = \frac{\Delta y}{\Delta x}<math>

(The delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change".)

Given two points (x₁, y₁) and (x₂, y₂), 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:

<math>m = \frac{y_2 - y_1}{x_2 - x_1}<math>

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 Δy is the "rise" and Δx 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(13,8) and Q(1,2). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line:

<math>m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}<math>

The slope is 1/2 = 0.5.

Example 2

If a line runs through the points (4, 15) and (3, 21) then:

<math>m = \frac{21 - 15}{3 - 4} = \frac{6}{-1} = -6<math>

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. The slope of a vertical line is not defined (it does not make sense to define it as +∞, because it might just as well be defined as -∞).

The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:

<math>m = \tan\,\theta<math>

and

<math>\theta = \arctan\,m<math>

(see trigonometry).

Two lines are parallel if and only if their slopes are equal or if they both are vertical and therefore undefined; they are perpendicular (i.e. they form a right angle) 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, etc., is: by the angle in degrees or by the slope in a percentage.

Algebra

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting said function. Therefore, if the equation of the line is given in the form

<math>y = mx + b \,<math>

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 (x₀, y₀) on the line are both known, then the equation of the line can be found using the point-slope formula:

<math>y - y_0 = m(x - x_0) \,<math>

For example, consider a line running through the points (2, 8) and (3, 20). This line has a slope, m, of (20 - 8) / (3 - 2) = 12. One can then write the line's equation, in point-slope form: y - 8 = 12(x - 2) = 12x - 24; or: y = 12x - 16.

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 said point, and is thus equal to the rate of change of the function at that point.

Why calculus is necessary

If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition,

<math>m = \frac{\Delta y}{\Delta x}<math>,

is the slope of a secant line to a curve. For a line, the secant between any two points is identical to the line itself; however, this is not the case for any other type of curve.

For example, the slope of the secant intersecting y = x² at (0,0) and (3,9) is m = (9 - 0) / (3 - 0) = 3 (which happens to be the slope of the tangent at, and only at, x = 1.5).

However, by moving the points used in the above formula closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve. It follows that the secant line is identical to the tangent line when Δy and Δx equal zero; however, this results in a slope of 0/0, which is an indeterminate form (see also division by zero). The concept of a limit is necessary to calculate this slope; the slope is the limit of Δy / Δx as Δy and Δx approach zero. However, Δx and Δy are interrelated such that it is sufficient to take the limit where only Δx approaches zero.

This limit is the derivative of y with respect to x. It may be written (in calculus notation) as dy/dx.

Related topics

• The gradient is a generalization of the concept slope for functions of more than one variable.da:Hældningstal

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