Graphs and Functions
Slope of a Line
Learning Objectives
By the end of this section, you will be able to:
- Find the slope of a line
- Graph a line given a point and the slope
- Graph a line using its slope and intercept
- Choose the most convenient method to graph a line
- Graph and interpret applications of slope–intercept
- Use slopes to identify parallel and perpendicular lines
Before you get started, take this readiness quiz.
Find the Slope of a Line
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.
In mathematics, the measure of the steepness of a line is called the slope of the line.
The concept of slope has many applications in the real world. In construction the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. and as you ski or jog down a hill, you definitely experience slope.
We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope is a rate of change. See (Figure).
The slope of a line is
The rise measures the vertical change and the run measures the horizontal change.
To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical.
To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run,
- Locate two points on the line whose coordinates are integers.
- Starting with one point, 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:
Find the slope of the line shown.
| Locate two points on the graph whose
coordinates are integers. |
and |
| Starting at sketch a right triangle to
as shown in this graph. |
|
| Count the rise— since it goes down, 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 y decreases by 2 units as x increases by 3 units. |
Find the slope of the line shown.
Find the slope of the line shown.
How do we find the slope of horizontal and vertical lines? To find the slope of the horizontal line, we could graph the line, find two points on it, and count the rise and the run. Let’s see what happens when we do this, as shown in the graph below.
Let’s also consider a vertical line, the line as shown in the graph.
The slope is undefined since division by zero is undefined. So we say that the slope of the vertic