How to Find Slope From a Graph?
The process of finding the slope from graph uses the slope formula rise/run. When the graph of a line is given and we are asked to find its equation, the first thing that we have to do is to find it slope. We can choose any two points (x₁, y₁) and (x₂, y₂) on the line and use the formula (y₂ - y₁) / (x₂ - x₁) for calculating the slope from a graph.
Let us see more ways of calculating slope from a graph along with more examples. Let us also learn how to find the slope of horizontal and vertical lines.
Finding Slope From a Graph
The slope of a line is the ratio of rise to run. Hence, here are the steps to find slope from a graph.
- Select any two random points on the graph of the line (preferably with integer coordinates).
- Label them as A and B (in any order).
- Calculate the "rise" from A to B. While going vertically from A to B, if we have to go
"up", then the rise is positive;
"down", then the rise is negative. - Calculate the "run" from A to B. While going horizontally from A to B, if we have to go
"right", then the run is positive;
"left", then the run is negative. - Now, use the formula: slope = rise/run.
Here is an example of the graph of a line.
Here, we have taken A = (1, 1) and B = (0, 3). Note that we have taken the points with integer coordinates here. Make a right-angled triangle starting from A and ending at B which makes the process of finding rise and run easier. Here, we have to move vertically "up" to reach B from A and hence rise = +2; and we have to move horizontally "left" to reach from A to B and hence run = -1. So slope = rise/run = 2/-1= -2.
We do not need to choose these points only to calculate the slope and also we do not need to choose them in the above order. Here you can see the graph of the same line where the same points are chosen in a different order and different points are chosen. Observe that the slope of the line (final answer) will be ultimately the same.
Calculating Slope From Graph Using Slope Formula
The slope formula is used to find the slope of a line that joins two points (x₁, y₁) and (x₂, y₂). Using this formula, the slope of the line is, m = (y₂ - y₁) / (x₂ - x₁). We can use the same formula to find the slope of a line from its graph also. For this:
- Pick any two points on the line from its graph.
- Represent them as (x₁, y₁) and (x₂, y₂) in any order.
- Apply the formula m = (y₂ - y₁) / (x₂ - x₁) to find the slope.
Example: Consider the above graphs. Let us choose two points (-1, 5) and (1, 1) on it. Now, let
- (x₁, y₁) = (-1, 5)
- (x₂, y₂) = (1, 1)
Slope, m = (y₂ - y₁) / (x₂ - x₁)
= (1 - 5) / (1 - (-1))
= -4/2
= -2
We had got the same slope (-2) when we calculated the slope using rise/run also.
Finding Slope of a Horizontal Line From a Graph
A horizontal line is a line that is parallel to the x-axis. So all points on a horizontal line have the same y-coordinates. Thus, for any two points on a horizontal line, the rise = 0. Hence, the slope = rise/run = 0/run = 0. Hence, the slope of a horizontal line is always 0. Here is an example.
Here, the slope of the horizontal line y = 3 is calculated using both methods rise/run and (y₂ - y₁) / (x₂ - x₁). You can notice that the slope in both cases is 0.
Finding Slope of a Vertical Line From a Graph
A vertical line is always parallel to the y-axis. Thus, all points on a vertical line have the same x-coordinates. Thus, for any two points on it, the run = 0. Hence, the slope = rise/run = rise/0 = undefined. Hence, the slope of a vertical line is always undefined. Here is an example.
Here, the slope of the vertical line x = 3 is calculated using both formulas rise/run and (y₂ - y₁) / (x₂ - x₁). You can notice that the slope in both cases is undefined.
Important Notes on Finding Slope From a Graph
- The slope from graph can be calculated by picking any two points on it and applying the formula rise/run.
- It can be also found by picking two points and applying the formula (y₂ - y₁) / (x₂ - x₁).
- The slope of a horizontal line is 0 always.
- The slope of a vertical line is undefined always.
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Slope From Graph Examples
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Example 1: Find the slope of the line from the following graph using the rise/run formula.
Solution:Let us take A = (-12, 0) and B = (0, -3). Then
- rise = -3 (as we have to come down vertically by 3 units from A to B)
- run = 12 (as we have to go right horizontally by 12 units from A to B)
So slope = rise/run = -3/12 = -1/4.
Answer: Slope from graph = -1/4.
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Example 2: A point on a line is (1, 4). Its slope is 2/3. Find any other point on the line by plotting the point and slope on the graph.
Solution:
The given point is (1, 4) and the slope is 2/3 = rise/ run.
Rise = +2 (so go vertically up by 2 units)
Run = +3 (go horizontally right by 3 units)
So we first plot the point (1, 4), rise it up by 2 units and move horizontally right by 3 units to get another point.
Thus, one point on the line can be (4, 6).
(If we write an equivalent fraction of slope, 2/3 = 4/6. In that case, another point on the line is (7, 8)).
Answer: A point on the line is (4, 6).
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Example 3: How to find slope from the following graph?
Solution:
The given line is vertical and hence its slope is undefined.
To explain this using rise/run, take any two points, say (3, 0) and (3, 3). Then rise = 3 and run = 0.
Then the slope = rise/run = 3/0 = undefined.
Answer: The slope of the line from given graph is undefined.
FAQs on How to Find Slope From a Graph
How Do You Find Slope From a Graph?
To find slope from a graph, find any two points on it. Preferably, select the lower point as A and the upper point as B. See how much we have to "rise" and "run" to go from A to B. Then the ratio rise/run would give us the slope (gradient). Note the following when you calculate the rise and run:
- rise is positive if you go up from A to B; it is negative if you go down.
- run is positive if you go right from A to B; it is negative if you go left.
How to Calculate the Slope of a Horizontal Line From a Graph?
We use the rise/run formula to calculate the slope from graph. For any two points on a horizontal line, the rise = 0 always. So its slope is 0/run = 0. So the slope of a horizontal line is always 0 and we can say it without calculating it.
How to Find Slope From Graph Using Slope Formula?
The slope (gradient) formula says the slope of a line with two points on it (x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁) / (x₂ - x₁). So when the graph of a line is given, just represent any two points on it as (x₁, y₁) and (x₂, y₂), find the difference in y-coordinates y₂ - y₁, find the difference in x-coordinates x₂ - x₁ and just divide them.
How to Use Rise/Run to Calculate Slope From a Graph?
Select any two random points on the line. Label them as A and B in any order. Calculate the rise (how much and in what direction we have to go vertically to reach from A to B) and run (how much and in what direction we have to go horizontally to reach from A to B). Then the fraction rise/run would give us the slope. Here the "direction" refers to the following.
- The direction is positive if we move "up" (rise) or " right" (run)
- The direction is negative we move "down" (rise) or " left" (run)
How to Find the Slope of a Vertical Line From a Graph?
The ratio rise/run formula is used to find the slope of any line. For any two points on a vertical line, the run = 0 always. So its slope is rise/0 = undefined. So we can say that the slope of a vertical line is always undefined without even calculating it.