
top
Chapter 12—Graphing 123 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. The term "change in" is often replaced with the Greek letter delta, D. Doing so, the ratio looks like this:  Examples Referring to line L_{1} 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 L_{1} is the ratio of Dy/Dx = 7/4 = 1.75.
 It is a negative slope. It slants downward.
Now referring to line L_{2}, the y dimension rises 4 units for every 2 units that x increases. So Dy = 4 and Dx = 2.  The slope of Line L_{2} is the ratio Dy/Dx = 4/2 = 2
 It is a positive slope. It slants upward.
 A visual inspection of Line L_{1} 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 L_{1} is 3. Referring to Line L_{2}, you can see that it rises 3 units for every unit of x. So the ratio is the same as forL_{1},and the slope is likewise 3. The slope of line L_{2} is 3. Note: Lines having the same slope (including the sign) are parallel lines.`  In this figure:  L_{1} has a slope of 1:4,or 0.25
 L_{2} 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 xaxis. 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 P_{1}, we can show its ordered pair as (x_{1},y_{1}); and the coordinates for a second point, P_{2}, would be (x_{2},y_{2}). 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 y_{2}  y_{1}, and the change in x (or Dx) is equal to x_{2}  x_{1}. Putting this information together, you should be able to see that: EQUATION Where:  m = the slope of a line plotted on the coordinate plane
 x_{1} and y_{1} are the coordinates for one point on the line
 x_{2} and y_{2} 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 (x_{1},y_{1}) and (x_{2}, y_{2}), 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. 
