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Chapter 12—Graphing

12-3 Finding the Slopes of Lines

 

Topics Covered in this Lesson

  • Introduction to the Slope of Lines
  • Given Two  Points,  Determine the Slope of a Line
  • Determining the Intercepts of a Line
  • Plotting a Line, Given Its Intercepts

Introduction to the Slope of Lines

Points are important parts of a line. It takes at least two points to define a line. You have already seen how two point determine where a line is located. But there is another very important quality:  the slope of the line. The slop is like the "slant" or "steepness" of a line on the coordinate plane.

  • A line with a positive slope rises from left to right.
  • A line with a negative slope falls from left to right.
  • A line with zero slop is a horizontal line.
  • A line with an infinite (undefined) slope is a vertical line.

Exercise

Use this exercises to make sure you can identify the type of slope of any line on a coordinate plane.

Definition

The slope of a line is defined as the ratio of the change in the y distance to the corresponding change in the x distance.fig120505.jpg (44434 bytes)

The term "change in" is often replaced with the Greek letter delta, D. Doing so, the ratio looks like this:

Examples

Referring to line L1 in this figure, you can see that the y dimension falls 7 units when the x dimension increases by 4. So Dy = -7 and Dx = 4.

The slope of Line L1  is the ratio of Dy/Dx = -7/4 = -1.75.
It  is a negative slope. It slants downward.

Now referring to line L2, the y dimension rises 4 units for every 2 units that x increases. So Dy = 4 and Dx = 2.

The slope of Line L2 is the ratio Dy/Dx = 4/2 = 2
It  is a positive slope. It slants upward.

A visual inspection of Line L1 shows that it rises ( y) 3 units for every unit of x. The ratio of the change in y to the change in x is 3:1 -- slope = 3/1 = 3.

The slope of line L1 is 3.

Referring to Line L2, you can see that it rises 3 units for every unit of x. So the ratio is the same as forL1,and the slope is  likewise 3.

The slope of line L2 is 3.

Note:
Lines  having the same slope (including the sign) are parallel lines.`


In this figure:

L1 has a slope of 1:4,or 0.25
L2 has a slope of -4:1, or -4

 

 

Exercise

Use this exercises to master the procedure for determining the slope of a line, given the values  for Dx and Dy.

Given Two  Points,  Determine the Slope of a Line

In the previous section, you saw how it is possible to determine the slope of a line by inspection -- by counting the number of units of change along the y- and x-axis. Quite often, however, you will find this "eyeball" procedure isn't accurate enough  for some applications. So we need a way to determine the slope of a line by calculation.

Consider that any point on the coordinate plane is represented by an ordered pair, (x,y).

Now consider how two points on the coordinate plane define a line. If we call  one of those points P1, we can show its ordered pair as  (x1,y1); and the coordinates for a second point, P2, would be (x2,y2).

You have already learned that the slope  of a line is given by: . This begins to take on special meaning when you understand that the change in y (or Dy) is equal to y2 - y1,  and the change in x (or Dx) is equal to  x2 - x1. Putting  this information together, you should be able to see that:

EQUATION

Where:

m = the slope of a line plotted on the coordinate plane
x1  and y1 are the coordinates for one point on the line
x2 and y2 are the coordinates for a second point on the line

 

Example

Suppose you are given these coordinates for two points on a coordinate plane: (4, 10) and (2, 4). Find the slope of the line drawn through them.

Step 1:  Assign the points.

Remember, it makes no difference which point you consider (x1,y1) and (x2, y2), just as long as you are consistent. So let's let (4,10) be point 1,  and (1,4) be point 2.

Step 2: Set up the equation and plug in the numerical values.

Step 3: Complete the math.

The slope is positive, so the line rises from left to right. The slope is 2, so the line rises 2 units upward for each unit to the right.

 

More Examples

 

 

 

 

Example 2:  Two points on a line have these coordinates:  (4,6) and (2, -8), Calculate the slope of the line.

Step1. Start with the equation

Step 2. Substitute the known values:

Step 3. Simplify:

So a line that includes points (4,6) and (2, -8) has a slope of 7.

Examples & Exercises

Use this exercises to master the procedure for determining the slope of a line, given the coordinates for two points on the line.

 

Something to Remember

  • A line with a positive slope rises from left to right (uphill)
  • A line with a negative slope rises from right to left (downhill)

Also: The value of the slope has no units. This is because it is the ratio of two terms having the same unit of measure.

David L. Heiserman, Editor

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All Rights Reserved

Revised: June 06, 2015