Gradient of a Line

Gradient of a Line is the measure of the inclination of the line with respect to the X-axis which is also called slope of a line. It is used to calculate the steepness of a line. Gradient is calculated by the ratio of the rate of change in y-axis to the change in x-axis.

In this article, we will discuss the gradient of a line, methods for its calculation, the gradient of a curve, applications of gradient of a line, some solved examples, and practice problems related to the gradient of a line.

What is Gradient?

Gradient refers to the rate of change of a quantity with respect to some independent variable.

In mathematics, gradient implies the degree of inclination of any entity towards something. Gradient also means the rate of descent or ascent of any hill or highway. Gradient, in mathematics, provides insight of the direction and steepness of the line.

What is Gradient of a Line?

Gradient of a line is the measure of its degree of inclination with respect to the X-axis. For gradient of a line, the term “slope” is used. In mathematics, the gradient of a line is typically denoted by the letter ‘m’.

Gradient of a line is calculated as the ratio change in y coordinate to the change in x coordinate of a line. It is also calculated as the trigonometric tangent of angle made by the line with positive X-direction in anticlockwise direction.

Gradient of a Line Formula

The formula for Gradient of a Line passing through two points (x_1, y₁) and (x₂, y₂) is given by,

m = (y₂ −y₁ )/(x₂ −x₁ )

OR

m = Δy/Δx

It represents the change in ordinates with respect to change in abscissa for a line. Various methods to calculate gradient or slope of a line are discussed as follows.

How to Calculate Gradient of a Line?

There are various methods to calculate gradient of a line, i.e. its degree of inclination with respect to X-axis. This degree of inclination is also defined in terms of trigonometric tangent of the angle made by the line with positive X-direction taken in anticlockwise direction.

Gradient of a line is particularly called the slope of a line. Different methods to calculate the gradient of a line based on type of inputs available are discussed as follows:

• Angle of Inclination
• Coordinates of two points on the line
• Equation of line

Angle of Inclination

Let the angle made by the line with positive X direction taken in anticlockwise direction be θ and m denote the gradient or slope of the line. Then we have,

m = tan θ

Hence, the gradient of a line can be measured by evaluating the value of tangent of the angle made by the line with positive X-direction taken in anticlockwise direction.

Coordinates of Two Points

Let (x₁, y₁) and (x₂, y₂) denote the coordinates of two points on the line and m be the slope of the line. Then we have,

m = (y₂-y₁)/(x₂-x₁)

Hence, the other method to calculate the gradient of a line is given the ratio of change in y-coordinate to the change in x-coordinate.

Equation of Line

If equation of the line is given as ax + by + c = 0, we represent it in the form y = mx + c, where ‘m’ is the slope or gradient of the line. It is mathematically expressed as follows,

We have ax + by + c = 0

⇒ y = (-a/b) x + (-c/b)

⇒ m = -a/b

Thus, for a line represented as ax + by + c=0, slope or gradient is given as m = -a/b, i.e. -(coefficient of x)/(coefficient of y).

Gradient of a Curve

Gradient of a curve is the gradient or slope of the tangent drawn to the curve at a given point. It determines the rate of change of the functional value represented by the curve at the given point.

Gradient of a Curve is not a constant value necessarily, i.e. gradient of a curve depends on the point where it needs to be calculated.

It is generally calculated as derivative of the function represented by the curve. For instance, a curve is represented as y = f(x), then gradient or slope of the tangent at any point is given as follows,

m = dy/dx = f ‘(x)

Note: A positive value of gradient indicates that the curve or function is increasing whereas negative value indicates that curve or function is decreasing. If the slope value is zero, it indicates that the function is constant.

Gradient of Different Lines

There can be various different lines that can be named such as:

• Horizontal Line
• Vertical Line
• Perpendicular Lines
• Parallel Lines

Let’s discuss the gradient of these lines one by one:

Gradient of Horizontal lines

Gradient of a horizontal line is zero as it is parallel to the X-axis, thereby making an angle of 0° with X-axis. As trigonometric tangent of 0 is zero, thus gradient of a horizontal line is zero.

It can also be defined by change in y-coordinate (which is 0 here) to the change in x-coordinate, which is represented as:

Gradient of Horizontal Line = 0/change in x-coordinate = 0

Gradient of Vertical lines

Gradient of a vertical line is infinite or not defined as it makes an angle of 90° with X-axis and trigonometric tangent of 90° being infinite or not defined, the slope of a vertical line is also not defined.

It can also be defined by change in y-coordinate to the change in x-coordinate (which is 0 here), which is represented as:

Gradient of Horizontal Line = Change in y-coordinate/0 = Not Defined

Gradient of a Line Perpendicular to Another Line

The slope of Perpendicular Lines are inversely proportional to each other and their product is -1. Let the gradient of a line = m₁ and gradient of the line perpendicular to it = m₂. Then, we have the mathematical relation m₁ × m₂ = -1. Thus, gradient of a line perpendicular to another line would be,

m₁ = -1/m₂ 

Gradient of Parallel Lines

The slope of Parallel Lines is the same as both the lines are at the same inclination with the positive x-axis. In other words, if the slope of one line is m then the slope of a line parallel to that line is also m.

Types of Gradient of a Line

Gradient of a line in general tells about the inclination of a line with respect to the X-axis. The types of gradient of a line include:

• Positive Gradient
• Negative Gradient
• Zero Gradient
• Undefined Gradient

Gradient can be used to tell about the extent of inclination of the line with respect to positive X-axis direction. It is discussed as follows:

Positive Gradient

Positive Gradient of a line implies that the line makes an acute angle with positive direction of X-axis, i.e. the angle between the line and positive X-axis taken in anticlockwise direction ranges from 0 degree to 90°.

For curves, if gradient of the tangent line drawn at a point is positive, then the curve is said to have an increasing nature.

Negative Gradient

Negative Gradient of a line implies that the line makes an obtuse angle with positive direction of X-axis, i.e. the angle between the line and positive X-axis taken in anticlockwise direction ranges from 90° to 180°.

For curves, if gradient of the tangent line drawn at a point is negative, then the curve is said to have an decreasing nature.

Zero Gradient

Zero gradient is defined for the horizontal lines only. If the line is horizontal (parallel to x-axis), the gradient is zero as tan 0°= 0. Hence, in this case, m = 0.

Undefined Gradient

The gradient of all vertical lines is always undefined. If the line is vertical (perpendicular to x-axis), the gradient is undefined as tan 90° = undefined.

Applications of Gradient of a Line

Gradient of a Line gives information about a line which can be used to determine its relation with other lines and understanding its geometry. It is useful in various engineering applications and holds importance in subjects such as coordinate geometry, vector algebra and three dimensional geometry. Some important applications of Gradient of a Line are listed as follows:

• It can be used to find equation of the line if a point on the line is known.
• It gives information about inclination of the line with respect to X-axis.
• Angle between two lines can be calculated with help of their gradient’s value.
• Gradient of a line can be used to check if the two lines are parallel or perpendicular to each other.
• Gradient of a curve helps to know about the increasing or decreasing nature of the curve.

Read More,

• Slope-Intercept form of straight lines
• Standard Form of a Straight Line
• Equation of a Straight Line

Solved Questions on Gradient of a Line

Question 1: Find the gradient of a line which passes through the points (3,5) and (1,4).

Solution:

We know that gradient of a line passing through (x₁, y₁) and (x₂, y₂) is given as,

m = (y₂-y₁)/(x₂-x₁)

Therefore, gradient or slope of the given line would be,

m = (5-4)/(3-1) = 1/2

Thus, we have calculated the gradient of given line as 1/2.

Question 2: A line makes an angle of 60° with positive X-direction in anticlockwise direction. Find the gradient of line.

Solution:

Here, we have θ = 60° and we know that,

Gradient of a line, m = tan 60° = √3

Thus, gradient of the given line is found to be √3.

Question 3: What is gradient of the line represented as 3x+4y+5=0?

Solution:

We know that,

Slope or Gradient of a line = -(coefficient of x)/(coefficient of y)

Therefore, for given line, we have,

Gradient, m = -3/4

Question 4: Determine the gradient and nature of the curve represented as y = x² + 5x + 12 at x=2.

Solution:

For any curve, gradient is the slope of tangent line drawn at the given point and it given as dy/dx. Here we have,

y = x² + 5x + 12

⇒ dy/dx = 2x + 5

At x = 2, dy/dx = 2 × 2 + 5 = 4 + 5 = 9

Thus, a positive value of gradient is obtained which indicates that the function is increasing at x = 2.

Practice Problems on Gradient of a Line

Problem 1: Find the gradient of a line passing through the points (1, 2) and (3, 4).

Problem 2: Find the angle made by a line with positive direction of X-axis whose gradient is 1/√3.

Problem 3: What is gradient of the line represented by the equation 4x+3y+12=0.

Problem 4: Find the expression for Gradient of the curve represented as y = ln x at any point.

Problem 5: Find the gradient of the curve y = sin x at x = π/2.

Gradient of a Line: FAQs

What is the meaning of gradient of a line?

The gradient of a line, often referred to simply as the slope, describes the steepness or inclination of the line with respect to the x-axis.

What does zero gradient of a line indicate?

Zero gradient of a line indicates that the line is a horizontal one, i.e. it is parellel to the X-axis.

What is meant by Gradient of a curve?

A curve’s gradient gives information about the rate of change of function with respect to the independent variable. A positive gradient indicates increasing nature of curve while a negative gradient implies the decreasing nature.

What is the formula to find gradient of a line passing through (x₁,y₁) and (x₂,y₂)?

The gradient of a line passing through the points (x₁,y₁) and (x₂,y₂) is given as,

m = (y₂-y₁)/(x₂-x₁)

What does ‘m’ indicate in equation of the line y = mx + c?

Gradient or slope of the line is indicated by ‘m’ in the equation of line y = mx + c.

What is gradient of a line which is parallel to the y-axis?

A line parallel to Y-axis is perpendicular to the X-axis, i.e. it makes an angle of 90° with the X-axis. Thus, gradient of such line would be tan 90° which is undefined and can be taken as infinity for calculations.