How To Find Slope on a Graph
- How to Find Slope of a Graph?
- Slope on a Graph
- Slope From a Graph Using Rise-Over-Run Approach
- Slope From Graph Using Slope Formula
- Slope of a Vertical Line From a Graph
- Slope of a Horizontal Line From a Graph
- Important Notes For Finding Slope From a Graph:
- Solved Examples
- Practice Problems
- Frequently Asked Questions
How to Find Slope of a Graph?
To find the slope of a straight line from a graph, you need to determine the change in the vertical (`y`-axis) direction divided by the change in the horizontal (`x`-axis) direction between two points on the graph. The slope represents the rate of change of the dependent variable `(y)` with respect to the independent variable `(x)`. For a straight line, slope helps us understand how steep or slant the straight line is.
Slope on a Graph
Here are the steps for finding the slope from a graph:
`1`. Select Two Points on the Line: Choose any two lattice points on the line represented by the graph. These points should have different `x` and `y` coordinates.
`2`. Identify the Coordinates: Determine the coordinates of the selected points. Let's denote the coordinates of the first point as \((x_1, y_1)\) and the coordinates of the second point as \((x_2, y_2)\).
`3`. Calculate the Change in `y` and Change in `x`: Find the difference between the `y`-coordinates and difference between the `x`-coordinates of the two points:
- Change in `y` (\(Δy\)) `=` \(y_2 - y_1\)
- Change in `x` (\(Δx\)) `=` \(x_2 - x_1\)
`4`. Calculate the Slope: Divide the change in `y` by the change in `x` to find the slope (\(m\)):
Slope `(m) = \frac{Δy}{Δx}`
Slope from a Graph Using Rise-Over-Run Approach
We can also find the slope of a graph by using the rise-over-run method. The rise-over-run approach is a straightforward method to find the slope of a line from a graph. Here's how you can do it:
`1`. Select Two Points on the Line: Choose any two distinct points on the line present on the graph. These points should have different `x` and `y` coordinates.
`2`. Calculate the Rise: Determine the vertical change between the two points. This is the difference between the `y`-coordinates of the two points. If you're moving from the first point to the second point and
- if the second point is above the first point, the rise is positive.
- if the second point is below the first point, the rise is negative.
`3`. Calculate the Run: Determine the horizontal change between the two points. This is the difference between the `x`-coordinates of the two points. If you're moving from the first point to the second point and
- if the second point is to the right of the first point, the run is positive.
- if the second point is to the left of the first point, the run is negative.
`4`. Find the Slope: Divide the rise by the run. The slope of the line is equal to the ratio of the vertical change (rise) to the horizontal change (run).
Here's the formula for finding the slope using the rise-over-run approach:
`\text{Slope} = \frac{\text{Vertical Change (Rise)}}{\text{Horizontal Change (Run)}}`
Example: Find the slope of the straight line shown in the graph below.
Solution:
We use the highlighted points `A` and `B` to find the slope.
Here the second point `A` is `2` units below the first point `B`. Hence rise `= -2`.
Next, we move `1` unit to the right. Hence run `= +1`
The formula for finding the slope using rise-over-run approach is
`\text{Slope} = \frac{\text{Vertical Change (Rise)}}{\text{Horizontal Change (Run)}}`
`\text{Slope} = \frac{-2}{1} = -2`
Therefore, the line has a negative slope. The slope value is `-2`.
The choice of points and their order doesn't affect the calculation of the slope, as long as the points are distinct and lie on the line. The slope of the line remains the same regardless of which points are chosen and the order in which they are selected.
This graph demonstrates the principle that the slope of a line is determined by its steepness, which remains constant throughout its length. The slope is solely dependent on the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line, regardless of their arrangement or coordinates.
Slope from Graph Using Slope Formula
Based on the rise-over-run approach, there is a slope formula which we can use to find the slope of a line that passes through the two points `(x_1, y_1)` and `(x_2, y_2)`. The slope formula is `m = (y_2 - y_1) / (x_2 - x_1)`, where `m` is the slope of the line. We can use the same formula to find the slope of a line from its graph also. For this:
`1`. Pick any two points on the line from its graph.
`2`. Represent them as `(x_1, y_1)` and `(x_2, y_2)` in any order.
`3`. Apply the formula `m = (y_2 - y_1) / (x_2 - x_1)` to find the slope.
This approach is intuitive and widely applicable, making it a fundamental tool in analyzing linear relationships represented graphically.
Example: Find the slope of the line graphed below.
Solution:
Lets select the `2` points `(-2,-1)` and `(3,2)` and apply the slope formula.
Here is the calculation:
`1`. \( (x_1, y_1) = (-2, -1) \)
`2`. \( (x_2, y_2) = (3, 2) \)
`3`. Using the slope formula:
`m = \frac{y_2 - y_1}{x_2 - x_1}`
`= \frac{2 - (-1)}{3 - (-2)}`
`= \frac{3}{5}`
Hence, the given line has a positive slope of `\frac{3}{5}`.
Slope of a Vertical Line from a Graph
For a vertical line, the slope is undefined because it doesn't have a horizontal change between any points on the line. Here's how you can understand and reason about the slope of a vertical line from its graph:
`1`. Identify Two Points on the Line: Select any two distinct points on the vertical line. Since the line is vertical, the `x`-coordinates of these points will be the same, while the `y`-coordinates may differ.
`2`. Calculate the Rise and Run: Because the line is vertical, there is no horizontal change (run) between the two points. Therefore, the run is zero. However, there will be a vertical change (rise) between the two points, which is the difference in their `y`-coordinates.
`3`. Apply the Slope Formula: Use the slope formula, which is `m = \frac{Δy}{Δx}`, to calculate the slope. Since the run (\(Δx\)) is zero, and division by zero is undefined, the slope of a vertical line is undefined.
In summary, the slope of a vertical line is undefined. This is because the line doesn't extend horizontally; instead, it rises or falls infinitely as it extends vertically.
Let’s see an example:
The slope of the vertical line `x = 3` is calculated using both formulas `(\text{rise})/(\text{run})` and slope formula, `m = (y_2 - y_1) / (x_2 - x_1)`. The slope in both cases is undefined.
Slope of a Horizontal Line from a Graph
When dealing with a horizontal line, the slope is zero because it doesn't have a vertical change between any two points on the line. Here's how you can find the slope of a horizontal line from its graph:
`1`. Identify Two Points on the Line: Select any two distinct points on the horizontal line. Since the line is horizontal, the `y`-coordinates of these points will be the same, while the `x`-coordinates may differ.
`2`. Calculate the Rise and Run: Because the line is horizontal, there is no vertical change (rise) between the two points. Therefore, the rise is zero. However, there will be a horizontal change (run) between the two points, which is the difference in their `x`-coordinates.
`3`. Apply the Slope Formula: Use the slope formula, which is `m = \frac{Δy}{Δx}`, to calculate the slope. Since the rise (\(Δy\)) is zero, the slope will be zero regardless of the value of \(Δx\).
In summary, the slope of a horizontal line is always zero. This is because the line doesn't rise or fall as it extends horizontally across the coordinate plane.
Let’s see an example:
The slope of the horizontal line `y = 3` is calculated using both methods `(\text{rise})/(\text{run})` and slope formula, `m = (y_2 - y_1) / (x_2 - x_1)`. It can be seen that the slope in both cases is `0`.
Important Notes for Finding Slope from a Graph
While finding the slope from a graph, there are several important notes to keep in mind:
`1`. Definition of Slope: Slope represents the rate of change between two points on a line. It indicates how steeply the line rises or falls as it extends from one point to another.
`2`. Slope Formula: The slope (\(m\)) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:
`m = \frac{Δy}{Δx} = \frac{y_2 - y_1}{x_2 - x_1}`
This formula is derived from the rise-over-run concept and measures the vertical change divided by the horizontal change between the two points.
`3`. Horizontal Lines: The slope of a horizontal line is always zero. This is because horizontal lines extend infinitely in the horizontal direction but do not rise or fall.
`4`. Vertical Lines: The slope of a vertical line is undefined. This is because vertical lines extend infinitely in the vertical direction but do not move horizontally.
`5`. Positive and Negative Slopes: A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right.
`6`. Zero Slope and Undefined Slope: A slope of zero indicates a horizontal line, while an undefined slope indicates a vertical line.
`7`. Calculating Slope: To calculate the slope from a graph, select two distinct points on the line, determine their coordinates, and use the slope formula to find the slope between those points.
`8`. Interpreting Slope: Slope provides valuable information about the direction and steepness of a line. It is widely used in various fields, including mathematics, physics, engineering, and economics.
Solved Examples
Example `1`: Find the slope of the line passing through the points \( A(-3, -5) \) and \( B(-3, 7) \).
Solution:
Since both points have the same `x`-coordinate, the line is vertical and has an undefined slope.
Example `2`: Find the slope of the line passing through the points \( A(-5, 3) \) and \( B(7, -1) \).
Solution:
Using the slope formula:
`m = \frac{-1 - 3}{7 - (-5)} = \frac{-4}{12} = -\frac{1}{3}`
So, the slope of the line passing through points \( A \) and \( B \) is `-\frac{1}{3}`.
Example `3`: A straight line passing through the point \( (2, 4) \) with a slope of `\frac{3}{2}`. If the `y`-coordinate of the second point on the line is `10`, what is the `x`-coordinate of that second point?
Solution:
We know the formula for the slope of a line is:
`m = \frac{y_2 - y_1}{x_2 - x_1}`
Given \( (x_1, y_1) = (2, 4) \) and `m = \frac{3}{2}`, we can plug these values into the formula to find \( x_2 \):
`\frac{3}{2} = \frac{10 - 4}{x_2 - 2}`
`\frac{3}{2} = \frac{6}{x_2 - 2}`
Now, cross multiply:
\( 3(x_2 - 2) = 2 \cdot 6 \)
\( 3x_2 - 6 = 12 \)
\( 3x_2 = 18 \)
\( x_2 = 6 \)
So, the `x`-coordinate of the second point is \( x_2 = 6 \).
Example `4`: Find the slope of the line shown in the below graph.
Solution:
Look at the graph and locate any two points like the points `(2,2)` and `(5,4)`.
To find the slope of the line passing through the points `(2,2)` and `(5,4)`, you can use the slope formula:
`m = \frac{y_2 - y_1}{x_2 - x_1}`
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
Let's use the points `(2,2)` and `(5,4)`:
`m = \frac{4 - 2}{5 - 2}`
`m = \frac{2}{3}`
So, the slope of the line passing through the points `(2,2)` and `(5,4)` is `\frac{2}{3}`.
Example `5`: Find the slope of the horizontal line passing through `y= 3`.
Solution:
The formula for slope (\( m \)) is given by:
`m = \frac{\text{change in } y}{\text{change in } x}`
For a horizontal line, the change in \( y \) is always `0` because the line doesn't rise or fall. So, if the line passes through \( y = 3 \), the value of \( y \) is always `3`. Hence, there's no change in \( y \) for any change in \( x \). Therefore, the slope (\( m \)) is:
`m = \frac{\text{change in } y}{\text{change in } x} = \frac{0}{\text{any value of } x} = 0`
So, the slope of a horizontal line passing through \( y = 3 \) is `0`.
Practice Problems
Q`1`. Find the slope of the line passing through the points \( A(2, 3) \) and \( B(5, 9) \).
- \( 2 \)
- \( -2 \)
- \( 5 \)
- \( -9 \)
Answer: a
Q`2`. Find the slope of the line shown in the below graph.
- \( -1 \)
- \( -2 \)
- \( 2 \)
- \( -4 \)
Answer: b
Q`3`. Find the slope of the line passing through the points \( A(3, 1) \) and \( B(-1, 5) \).
- \( 3 \)
- \( -2 \)
- \( -1 \)
- \( 1 \)
Answer: c
Q`4`. Find the slope of the line shown below.
- \( -1 \)
- \( 1 \)
- \( -2 \)
- undefined slope
Answer: d
Q`5`. Find the slope of the line passing through the points \( A(1, 4) \) and \( B(3, 4) \).
- \( 0 \)
- \( 1 \)
- \( -1 \)
- \( 2 \)
Answer: a
Frequently Asked Questions
Q`1`. What is slope, and why is it important?
Answer: Slope measures the steepness of a line and represents the rate of change between two points. It's crucial for understanding the direction and magnitude of change in various contexts, such as physics, engineering, economics, and more.
Q`2`. How do you find the slope from a graph?
Answer: To find the slope from a graph, select two distinct points on the line, calculate the difference in their `y`-coordinates (rise) and `x`-coordinates (run), and then divide the rise by the run. This can also be done using the slope formula: `m = \frac{Δy}{Δx}`.
Q`3`. What does a positive slope indicate?
Answer: A positive slope indicates that the line rises from left to right. It means that as the `x`-values increase, the `y`-values also increase.
Q`4`. What does a negative slope indicate?
Answer: A negative slope indicates that the line falls from left to right. It means that as the `x`-values increase, the `y`-values decrease.
Q`5`. What does a slope of zero mean?
Answer: A slope of zero indicates a horizontal line. This means that the line neither rises nor falls as it extends.
Q`6`. What does an undefined slope mean?
Answer: An undefined slope indicates a vertical line. This means that the line rises or falls infinitely as it extends vertically.
Q`7`. Can any two points on a line be used to find its slope?
Answer: Yes, any two distinct points on a line can be used to find its slope. However, it's important to choose points that are easy to read accurately from the graph.
Q`8`. How is slope used in real-world applications?
Answer: Slope is used in various real-world applications, such as calculating gradients in terrain, determining rates of change in economics and finance, analyzing motion in physics, and more.
Q`9`. Can a line have a slope of infinity?
Answer: Yes, a vertical line has an undefined slope, which can be interpreted as infinity. This occurs because the run between any two points on a vertical line is zero, leading to division by zero in the slope formula.