How to Find the Slope of a Line From Its Graph

Slope of a line is the measure of its inclination with the positive x-axis. Mathematically, the slope is defined as “rise over run”. The slope is the measure of the inclination or slant of the line, whether it is rising or falling.

In this article, we will be discussing how to calculate the slope of a line on a graph, with an easy-to-understand formula, along with examples and others.

How to Find Slope From a Graph?

• Imagine a line on a graph. The slope of the line can be determined by observing how much it slants from left to right. If the line has a significant slant, it indicates a higher slope, which means there is a considerable difference between its side-to-side position (the x-value) and its up-and-down position (the y-value). Conversely, if the line doesn’t slope considerably, it has a smaller slope, indicating that there is only a little change in its up-and-down position for every unit change in its side-to-side position.

• To determine the slope of a line, a formula is used. Here’s how it works: You need to choose two points on the line, such as (x₁, y₁) and (x₂, y₂). Next, input these points into the following formula,

m = (Y₂ – Y₁) /(X₂ – X₁)

This formula will give you the slope of the line, which is represented by the variable ‘m’ and is calculated as (y₂ – y₁) / (x₂ – x₁).

Slope of a Line

• Slope is determined by dividing the difference between the two points’ changes in x and y values (x₂ and x₁) by the change in y values (y₂ – y₁).

Finding Slope From a Graph

Slope of any line can easily be calculated using above formula as shown in the example added below:

Let’s say a line crosses points (2, 3) and (5, 7).

Use the above formula to find the slope (m):

m = (7 – 3) / (5 – 2)

m = 4 / 3

Slope in this particular case is 4/3, which means that the line is slanting upwards as you move right. We can choose any two separate points on the line, except for the vertical lines, to use in the slope formula.

The direction of the line’s slant is indicated by the slope’s sign (+ or -). A positive slope means that the line moves up as you travel right, whereas a negative slope makes the line move down.

Calculating Slope From Graph Using Slope Formula

Follow the steps added below to calculate slope from graph using slope formula,

• Choose two spots on the line. Choose two different locations on the line that are visible on the graph (please avoid those points that form a vertical line). Check whether you have calculated their coordinates accurately; or you x and y values.

• Call the first pair of points “(x₁, y₁)” and the second “(x₃, y₃).” In any case, it is immaterial which point you choose as (x₁, y₁), and (x₂, y₂) is the other one.

• Upon entering these numbers, perform necessary calculations using line formula below to derive for x and y values corresponding to the points you have designated: m = (y₂ – y₁) / (x₃ – x₁).

• Find the slope (m): Subsequently, observe the order of operations when executing division and subtraction steps (PEMDAS).

Example: Examine a line that intersects points (2, 3) and (5, 7). Apply the slope formula to determine the slope (m).

Solution:

Points are as follows: (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 7)

Equation(formula): m = (x₂ – x₁) / (y₃ – y₁)

To compute: m = (7 – 3) / (5 – 2) = 4 / 3

As a result, the line has a 4/3 slope. This shows a positive slope, which means that as you travel from left to right, the line slants upward.

Finding Slope of a Horizontal Line From a Graph

• Horizontal lines are unique because they move in the x-direction with no deviation. So, regardless of how far you go left or right, the up-and-down position (y-value) of the line does not change.

• When we are discussing slope, we are referring to the amount the line slants. However, if the line is horizontal, it is not tilted at all—it is a flat surface.

• When you employ the slope formula for a horizontal line, you are essentially dividing the difference in y-values over time. Nevertheless, since nothing changes in the y-values for a horizontal line, they remain constant. This leads to a result of zero when you subtract two equal values being subtracted from each other.

• For any horizontal line, it changes 0 in the y direction because the y-value is 0. This automatically gives the gradient of a horizontal line as 0 since division of any number by 0 equals 0.

Why Slope of a Horizontal Line is 0?

• In the context of a horizontal line, the rate of change in y-value is zero; however, the change in x-value may vary (x₂ – x₁). Therefore, the formula for calculating the slope becomes: m = 0 / (x₂ – x₁).

• Theoretically, division by zero is undefined. However, if you look at slope it means that no matter how far one moves in either direction (x), he/she remains at the same height which is what constitutes such a movement that doesn’t lead upwards or downwards called horizontal.

• A slope of zero therefore signifies that there is never any increment made when moving horizontally: y₂ – y₁/x₂ – x₁ = 0.

Example: Imagine that the points (3, 4) are crossed by a horizontal line. Its y-value (4) will not change no matter where you go on the line since it is horizontal. This would be the slope (m).

Solution:

For any non-zero value,

m = (4 – 4) / (x₂ – x₁)

m = 0

Essentially, the slope of every horizontal line, regardless of its x-coordinate, is always zero.

Finding Slope of a Vertical Line From a Graph

• Horizontal and vertical lines differ in some ways. The y-axis is followed by them, running straight down and up. When you gaze anywhere along a vertical line’s path, it is clear that all along that path from side-to-side location rotting constant stays the x-value.

• Slope is determined by the steepness of a line. Imagine, a vertical line simply goes up or down without slanting, while a slope does not exist. Moreover, in the context of a vertical line, the x-value is maintained. It is worth noting that a constant x-value subtracted from another always gives a zero.

Understanding Reason

“Slope formula relies on the change in both x and y values between two points on the line. m = (y₂ – y₁) / (x₂ – x₁)”

Why Slope of a Vertical Line is undefined?

All the points that define a vertical line have constant x-values. For this reason, x 2-x 1 is equal to zero in any formula. Every point on a vertical line must have the same x-value. This means that when we divide (x₂ – x₁), it becomes zero. Slope formula is undefined due to division by zero, if the formula has a zero denominator (x₂ – x₁ equals zero) then its slope is undefined.

Example: Consider a vertical line passing through the point (2, 5).

Since it’s vertical, its x-value (2) stays constant.

The slope (m) would be:

m = (y₂ – y₁) / (2 – 2) = (any value) / 0 = undefined

Slope From Graph Examples

Example: A line passes through points (1, 2) and (4, 6).

Using the formula:

m = (y₂ – y₁) / (x₂ – x₁)

• Points: (x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 6)

m = (6 – 2) / (4 – 1)

= 4 / 3

Slope (m) is 4/3, which is positive. This indicates that the line slants upwards as you move from left to right.

Example: Consider a line that goes through points (-2, 5) and (1, 1).

Using the formula:

m = (y₂ – y₁) / (x₂ – x₁)

• Points: (x₁, y₁) = (-2, 5) and (x₂, y₂) = (1, 1)

m = (1 – 5) / (1 – (-2))

= -4 / 3

Slope (m) is -4/3, which is negative. This signifies that the line slants downwards as you move from left to right.

Problems on Finding Slope of a Line

Problem 1: Imagine a horizontal line passing through the point (3, 4). Since it’s horizontal, its y-value (4) will remain constant.

Solution:

Using formula: m = (y₂ – y₁) / (x₂ – x₁)

Because it’s oriented horizontally, the difference in y-values (y₂ – y₁) is always zero (4-4)

In other words, the slope (m) is always zero irrespective of the x-value

Problem 2: A line on a graph passes through the points (2,1) and (5,4). Find the slope (m) of this line.

Solution:

Identify Coordinates: We are given two points on the line: (2, 1) and (5, 4)

Assign variables: Let’ (2, 1) as (x₁, y₁) and (5, 4) as (x₂, y₂).

Apply the slope formula: m = (y₂ – y₁) / (x₂ – x₁)

Substitute values: m = (4 – 1) / (5 – 2) = 3 / 3

Slope (m) of the line is 1

This indicates a positive slope, meaning the line slants upwards as you move from left to right.

Problem 3: Imagine a horizontal line on a graph that goes through the point (7, -2). What is the slope (m) of this line?

Solution:

Slope of a horizontal line (m) is 0.

Read More,

• Slope Formula
• Parallel Lines
• Slope-Intercept Form of a Line

FAQs on Finding Slope of a Line

What is Slope of a Line?

the measure of steepness or inclination of the line is called the slope of a line.

What is the formula to find the Slope of a tangent formula?

Slope of a line formula is,

Slope = m = tan θ = (y₂ – y₁)/(x₂ – x₁)

How to interpret the sign of slope?

• Positive Slope(+): Greater the positive slope, the steeper the upward slant of the line moving from left to right; it’s like climbing upstairs.
• Negative slopes, is like going downstairs/negative (-): The more negative it becomes, the steeper is this line when going down/right on the graph paper.
• Zero slope (0) is illustrated by drawing horizontal lines where there is no change from point-to-point in terms of y-value as it varies along x-axis.

How is the concept of slope used in real-world applications?

Slopes can be used in various fields:

• Motion: Movement entails thinking about a car in motion. Time is tracked while it changes position; resultant rate of change is used in determining speed or acceleration. This describes how much position changes over time.

• Rates of Change: How does the effective rate of growth change over time? If a city’s population grows each year, what is the rate coefficient? And the rate parameters determines how fast the value increases, too.

• Linear Relationships: If we think about how much rain there is, we will be able to tell how high the river will get because it shows an amount of increase in the river level for each inch of rain. In other words, slope may be likened unto key that explains connections between two different phenomena.