Slope tells us how steep a line is and what direction it is going on a graph. Mathematically, slope is usually defined as the change in y divided by the change in x. This is also called the rise over run.
Quick Answer: What are the 4 types of slope?
- Positive Slope
- Negative Slope
- Zero Slope
- Undefined Slope
What is Slope?
The slope is a number the defines how steep a line is. The larger the number, the closer it is to vertical. Also, a positive slope means it is getting higher from left to right. A negative slope means it is going doing from left to right.
Slope Formula
The formula for calculating slope needs 2 points. Take the x and the y value from each point to calculate the slope.
Slope = (y₂ − y₁) / (x₂ − x₁)
4 Types of Slopes and Examples
There are 4 different types of slopes, and all of them can be easily identified by their formula.
Positive Slope
A line with a positive slope goes up as you go from left to right
Example: y = 3x + 3
Negative Slope
A negative sloping line goes down from left to right
Example: y = -2x + 1
Zero Slope
A line with zero slope is a horizonal line, because it does not go up or down. Notice how “x” is not in the formula.
Example: y = 8
Undefined Slope
A line with an undefined slope is a perfectly vertical line. Notice this equation starts with an “x” and has no “y”
Example: x = 4
| Slope Type | Left to Right | Hint to Find |
|---|---|---|
| Positive | Up | Positive before x |
| Negative | Down | Negative before x |
| Zero | Flat | No x |
| Undefined | Vertical | No y |
How Do You Find the Slope of a Line?
There are 3 main ways to find the slope. Each depends on the information you are given
Method 1: Use the Slope Formula
If you know two points and their coordinates (x₁, y₁) and (x₂, y₂) you can use the slope formula m = (y₂ − y₁) / (x₂ − x₁)
Example: Points (1,2) and (5,10)
Slope = (10 – 2) / (5 – 1)
Slope = (8) / (4)
Slope = 2
Method 2: Rise Over Run
If you have two points from a graph, you can count the number of units up/down, and the units left/right. The slops is the rise/run
Method 3: Use an Equation
If an equation is given, you can find the slope in the equation
y = mx +b
Examples:
y = 4x + 3
The slope is 4
y = -2x + 6
The slope is -2
Common Mistakes Calculating Slope
Avoid the common mistakes when calculating the slope of a line
- Forgetting parentheses. Always use (y₂ − y₁)/(x₂ − x₁)
- Mixing up point 1 and 2. Double check the fomula
- Calling a vertical slope ‘infinite’. It is undefined.
More Advanced Details on Slopes
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 introduce the concept of slopes, we need 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. 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 all x-coordinates and only one possible 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
- Slope of a line is the measure of its steepness.
- The line’s angle with the x-axis gives the line’s steepness.
- Depending on the line drawn, the slopes are of 4 types. They are positive slope, negative slope, zero slope, and infinite slope.
- We can calculate the slope of a line using the formula m=y2-y1/x2-x1 or rise/run.
- We can also calculate the slope by finding the tangent of the angle formed between the line and the x-axis.
