Finding the Slope of a Line from its Graph
Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.
To find the slope, we must count out the rise and the run. But where do we start?
We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
Example
Find the slope of the line shown:
Solution
Locate two points on the graph, choosing points whose coordinates are integers. We will use \(\left(0,-3\right)\) and \(\left(5,1\right).\)
Starting with the point on the left, \(\left(0,-3\right),\) sketch a right triangle, going from the first point to the second point, \(\left(5,1\right).\)
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| Count the rise on the vertical leg of the triangle. | The rise is 4 units. |
| Count the run on the horizontal leg. | The run is 5 units. |
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values of the rise and run. | \(m=\frac{4}{5}\) |
| The slope of the line is \(\frac{4}{5}\). |
Notice that the slope is positive since the line slants upward from left to right.
How to Find the slope from a graph.
- Locate two points on the line whose coordinates are integers.
- Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- Take the ratio of rise to run to find the slope. \(m=\frac{\text{rise}}{\text{run}}\)
Example
Find the slope of the line shown:
Solution
Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use (0, 5) and (3, 3). Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
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| Count the rise – it is negative. | The rise is −2. |
| Count the run. | The run is 3. |
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values of the rise and run. | \(m=\frac{-2}{3}\) |
| Simplify. | \(m=-\frac{2}{3}\) |
| The slope of the line is \(-\frac{2}{3}.\) |
Notice that the slope is negative since the line slants downward from left to right.
What if we had chosen different points? Let’s find the slope of the line again, this time using different points. We will use the points \(\left(-3,7\right)\) and \(\left(6,1\right).\)
Starting at \(\left(-3,7\right),\) sketch a right triangle to \(\left(6,1\right).\)
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| Count the rise. | The rise is −6. |
| Count the run. | The run is 9. |
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values of the rise and run. | \(m=\frac{-6}{9}\) |
| Simplify the fraction. | \(m=-\frac{2}{3}\) |
| The slope of the line is \(-\frac{2}{3}.\) |
It does not matter which points you use—the slope of the line is always the same. The slope of a line is constant!
The lines in the previous examples had \(y\)-intercepts with integer values, so it was convenient to use the y-intercept as one of the points we used to find the slope. In the next example, the \(y\)-intercept is a fraction. The calculations are easier if we use two points with integer coordinates.
Example
Find the slope of the line shown:
Solution
| Locate two points on the graph whose coordinates are integers. | \(\left(2,3\right)\) and \(\left(7,6\right)\) |
| Which point is on the left? | \(\left(2,3\right)\) |
| Starting at \(\left(2,3\right)\), sketch a right angle to \(\left(7,6\right)\) as shown below. |
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| Count the rise. | The rise is 3. |
| Count the run. | The run is 5. |
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values of the rise and run. | \(m=\frac{3}{5}\) |
| The slope of the line is \(\frac{3}{5}.\) |