3. Equations and their Graphs
3.4 Understand Slope of a Line – optional
Learning Objectives
By the end of this section it is expected that you will be able to:
- Use to find the slope of a line from its graph
- Find the slope of horizontal and vertical lines
- Use the slope formula to find the slope of a line between two points
- Graph a line given a point and the slope
- Solve slope applications
When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?
In mathematics, the ‘tilt’ of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.
In this section, we will explore the concept of slope.
The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter .
Slope of a line
The slope of a line of a line is .
The rise measures the vertical change and the run measures the horizontal change between two points on the line.
Positive and negative slopes
We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line is going up; it has positive slope. The line is going down; it has negative slope.
Use to Find the Slope of a Line from its Graph
We’ll look at some graphs on the -coordinate plane and see how to find their slopes.
To find the slope, we must count out the rise and the run. But where do we start?
We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
EXAMPLE 1
Find the slope of the line shown.
TRY IT 1
Find the slope of the line shown.
Show answer
HOW TO: Find the slope of a line from its graph using m = rise / run.
- 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, .
EXAMPLE 2
Find the slope of the line shown.
| Locate two points on the graph whose coordinates are integers. | and |
| Which point is on the left? | |
| Starting at , sketch a right triangle to . | |
| Count the rise—it is negative. | The rise is . |
| Count the run. | The run is 3. |
| Use the slope formula. | |
| Substitute the values of the rise and run. | |
| Simplify. | |
| The slope of the line is . |
So increases by 3 units as decreases by 2 units.
What if we used the points and to find the slope of the line?
The rise would be and the run would be 9. Then , and that simplifies to . Remember, i