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4 Different Types of Slopes – How to find it?

Slope is a concept that comes into the picture when we talk about lines. To understand slopes, we need to know about lines. A line is a collection of points connected successively. It has a straight one-dimensional build that extends indefinitely on both sides.

To bring about the concept of slopes, we have to introduce a system of points. This system is the coordinate system. This system can be one-dimensional, two-dimensional, or three-dimensional. 

In a one-dimensional coordinate system, there is only one axis. It is called the x-axis. In a two-dimensional coordinate system, there are two axes. One axis lies horizontally, and the other one lies vertically. The horizontal axis or the reference axis is the x-axis. The vertical axis is the y-axis. Lastly, we have the three-dimensional coordinate system, which has three axes. These are the x, y, and z-axis, respectively. All the three axes lie perpendicular to each other.

The two-dimensional system is the most commonly used system. We can use this system for understanding slopes.

Slopes

A slope measures a line’s direction and steepness about a fixed axis.

Let us understand this definition by breaking it down.

When we draw a line on a cartesian coordinate system, we draw it for either one of the coordinate axes. This could be the x-axis or the y-axis. Alas, we measure slopes with the x-axis as a reference. Now, when drawing a line that intersects the x-axis, we need to consider the angle at which it intersects. This angle helps us understand the steepness of the line.

Types of slopes

The slope of a line can be of 4 types depending on the change in the x and y-coordinates. We can also classify them based on the angle they make with the x-axis.

Positive Slope

A line has a positive slope when both the x and y-coordinates successively increase or decrease. This means that if the x-coordinate increases, so do the y-coordinate and vice versa. From an angle perspective, if the angle is an acute angle, the slope will be positive.

Negative Slope

The relation between the x and y-coordinates reverses in a negative slope. This goes to say that when either one of the coordinates increases, the other will decrease. Thus, if the x-coordinate increases, the y-coordinate will decrease and vice versa. If the angle formed is obtuse, then the slope is negative.

Zero Slope

A perfectly horizontal line has zero slope. This type of line will be parallel to the x-axis. Such a line will have no x-coordinate and only be represented by its y-coordinate. In terms of its slope, the angle it is inclined at is zero as it moves along the x-axis.

Undefined Slope or Infinite Slope

Converse to a line with zero slope; an infinite slope is perfectly vertical. Such a line will be parallel to the y-axis. Hence, it will only be represented by its x-coordinate. The angle this line would make with the x-axis is 90 degrees.  

How do you find the Slope of a line?

As the slope is a measure of the steepness of a line, we can find the slope by simply finding the tangent of the angle at which the line interests the x-axis. It is given by,

m = tan(theta)

Where,
m= Slope of the given line
theta= Angle that the line makes with the x-axis

The simplest way to find a slope is by marking and naming any two points on the slope. Find the difference between the measures of the y-coordinates and repeat the same for the x-coordinates. Now, divide the differences in the y-coordinates to that of the x-coordinates. This value will give you the slope of the line drawn. This can be represented in equation form as,

m = y2 -y1/x2-x1
Or
m=rise/run

Where,
m= Slope of the given line
Or where,
rise= The number of units moved up or down counted from point to point
run= The number of units  moved left or right counted from point to point

If the count is upwards, the rise will be positive and vice versa. If the count in the run is towards the right, then it is positive. Otherwise, it is negative.

Summary

1. Slope of a line is the measure of its steepness.
2. The line’s angle with the x-axis gives the line’s steepness.
3. Depending on the line drawn, the slopes are of 4 types. They are positive slope, negative slope, zero slope, and infinite slope.
4. We can calculate the slope of a line using the formula m=y2-y1/x2-x1 or rise/run.
5. We can also calculate the slope by finding the tangent of the angle formed between the line and the x-axis.