What is slope? Explained with its types and calculations

In calculus, the slope of a line is a measure of how steeply it rises or runs. Specifically, it is clear as the ratio of the vertical change (rise) between two points on the line to the horizontal change (run) between those similar points.

The slope is typically represented by the letter “m” and can be positive, negative, or zero depending on the direction and steepness of the line. A positive slope means that the line is rising as it moves from left to right, while a negative slope means that the line is falling.

The concept of slope is important in many areas of mathematics and science, as it can be used to describe the rate of change of a function, the velocity of an object, or the gradient of a surface among other things.

So, while slopes may be associated with physical landscapes and they are also a fundamental concept in the world of mathematics and science to help us understand the behaviour of the world around us. In this article, we will discuss the basic definition, formula, and types.

What is the slope?

The slope of a line denotes the steepness of the line and is defined as the change in y divided by the change in x. It can also be defined as the ratio of the vertical change to the horizontal change between any two locations on the line. This definition is usually used in mathematics to evaluate the slope of a line in a coordinate plane.

Mathematical representation:

The formula of slope is,

m = slope = (∆y) / (∆x) = (y₂ – y₁) / (x₂ – x₁)

Where

(x₁, y₁) and (x₂, y₂) are two points on a line. The slope denotes the change in the y-axis divided by the change in the x-axis between these two points.

This formula can also be denoted as

slope = (rise) / (run)

Where

• “rise” denotes the change in the y-axis
• “run” denotes the change in the x-axis.

According to the line’s direction and steepness, the slope can be positive, negative, zero, or undefinable.

How to calculate the slope?

There are three steps for finding the slope of a straight line.

• Classify two points on the line.
• Choice one to be (x₁, y₁) and the further to be (x₂, y₂).
• Procedure to the slope formula to evaluate the slope.

Example:

Let’s about we have two points on a line (4, 8) and (6, 22).

Solution:

By applying the formula, we can determine the line’s slope.

m = Slope = (22 – 8) / (6 – 4)

m = Slope = 14 / 2

m = Slope = 7

Hence, the slope of the line passing through points (4, 8) and (6, 22) is 7.

The steepness of the line can also be measured with the help of a slope finder to get rid of time-consuming calculations.

Different types of slopes

We can identify the slope in different methods depending upon the connection between the two variables x and y and thus the value of the gradient or slope of the line attained. There are four different types of slopes given as

1. Positive slope
2. Negative slope
3. Zero slope
4. Undefined Slope

Positive Slope

A positive slope, or upward incline, is present when a line is viewed from left to right with the help of the formulas m = (y₂ – y₁) / (x₂ – x₁) = tan θ = f'(x) = dy/dx, one may determine a line’s positive slope. The presence of a positive slope indicates simultaneous increases or decreases in the two quantities represented along the two axes of the coordinate system. The rise-to-run ratio of the line has a positive value and a positive slope.

Negative Slope

A line that slopes downward from left to right is said to have a negative slope. When a line has a negative slope, the rise-to-run ratio is also negative. The formula for calculating it is m = (y₂ – y₁)/ (x₂ – x₁) = tan θ = f'(x) = dy/dx. If one quantity is falling and another is increasing, the slope is negative.

Zero Slope

The x-axis of the coordinate structure runs straight to a line with zero slopes. The line with a zero slope provides an angle of 0° or 180° for the upward x-axis directions. For any two points along a line with a slope of none, the value of the y-axis is constant. The point where the line with zero slopes connects the y-axis is (0, a) which is “a” units away from the x-axis.

Undefined Slope

The undefined slope is comparable to the slope of a vertical line. The x-coordinates are independent of the value of the y-coordinates. The vertical lines don’t run left or right instead they rise straight up or fall straight down. The slope is determined by dividing the change in y coordinates by the change in x coordinates. Since a vertical line’s x coordinates do not change and the slope is undefined or impossible because the denominator is 0.

Calculation:

Example 1:

Evaluate the slope of a line between the points (7, –4) and (9, 5).

Solution:

Step 1:

Assumed, the points (7, –4) and (9, 5).

Step 2:

As per the slope formula, we know that,

m = (y₂ – y₁) / (x₂ – x₁)

 Step 3:

m = (5-(-4)) / (9-7)

m = 9/2

Example 2:

Evaluate the slope of a line between (–6, 3) and (5, –2).

Solution:

Step 1:

Given, (–6, 3) and (5, –2) are the two points.

Step 2:

As per the slope formula, we know that,

m = (y₂ – y₁) / (x₂ – x₁)

Step 3:

So, the slope of the line,

m = (-2- 3) / (5- (-6))

m = -5/11

Frequently asked questions:

Question 1:

What is the introduction of the slope?

Solution:

The slope of a line expresses to us how vertical it is. The slope develops steeper as it gets bigger. As the slope gets smaller, it flattens out. Slope can also be used to determine if a line is horizontal, vertical, or moving upward or downward.

Question 2:

Why is it called slope?

Solution:

According to linguists, the term slope is a derivative of the Middle English adverb a slope, which means “at an angle.” You can use the word’s noun form to refer to anything that is angled or on a slope, such as a steep hill or a parking garage ramp.

Conclusion:

In this article, we have discussed the slope, the formula, how to calculate the slope, important techniques, and its types. Moreover, discussed how to evaluate the slope with the help of examples. The slope is a most common topic, hope you can solve the easily related problem by reading of this article.