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Investigating Slopes of Lines
Often when a student hears the word slope, many different things come to mind. Some will say “rise over run”, others say “change in y over change in x”, some will give a formula and still others say slope can be determined from the equation of the line. What can be confusing is that all of these descriptions are correct.

Let’s look at slope from these various points of view.

What Is Slope?
Slope is a characteristic of a line and can either be positive or negative.

This graph has a positive slope.



When you look at the line going from left to right, it appears as if you are going up a hill. Another way to describe what is happening is to say that as the x-values increase the y-values also increase. Whenever both the x and y variable change in the same direction, the slope of the line will be positive.

The graph below has a negative slope.



When you look at the line going from left to right, it appears as if you are going down a hill. You can also say that as the x-values are getting larger, the y-values are getting smaller. When the x and the y variable change in opposite directions, the slope will be negative.
How Can I Find Slope?

The answer to this question depends on the type of information you are given.
  1. Given A Graph
As seen from the previous graphs, you can make a determination about the slope of a line simply by looking at a graph. Depending on the amount of detail shown on the graph, it may be possible to find an exact value for the slope or you may only be able to tell whether the slope is positive or negative.

If you cannot determine particular points on the graph, you will only be able to describe slope as being positive or negative.

Let's Practice:
  1. Is the slope of the graph below positive or negative?




The slope of this graph is negative. As we move along the line from left to right, we are going down a hill which means the slope is negative.
  1. Is the slope of the graph below positive or negative?


The slope of this graph is positive. As we move along the line from left to right, we are going up a hill which means the slope is positive.
When it is possible to read two points from a graph, you can find an exact value for slope. To see how this is done go to the section “Given Two Points” later in the lesson.
  1. Given An Equation
Equations of lines can be presented in many forms. The easiest form to work with for finding slope is called the slope-intercept form of a line and is written as . In this form, m is the slope and b is the y-intercept. In this lesson, we will only talk about the slope. For more information on the y-intercept, click here. (link to linear equations intercepts.doc).

More Practice:
  1. Find the slope of y = -2x + 5.
When you are given an equation in slope-intercept form is given, you can quickly determine that the slope of that line is -2.
When a line is given in a different form, it requires a little more work to find the slope. Sometimes lines are given in general form, which is written as . This form is useful for finding intercepts (another link to linear equations intercepts.doc), but makes finding slope a little harder than the slope-intercept form. Since finding slope from the slope-intercept form of a line is so easy, you should change the general form into slope-intercept form to find the slope.
  1. Find the slope of 2x + 3y - 5 = 0
Changing this into slope-intercept form requires getting y on one side by itself and everything else on the other side of the equation.
From this form, you can determine that the slope is .
When given the general form , some students like to just find the slope by using . However, when you begin using the slope formula discussed in the next section, it can become confusing to have too many formulas for slope.


If you are given an equation in any other form, you should put it in slope-intercept form to find the slope.
  1. Given Two Points
Suppose you know that a line goes through the points (-1, 3) and (4, -2). You do not have a graph (although you could draw one) and you do not have the equation of the line that goes through those points. In this situation, you need to know the formula for the slope of a line.

If you are given two points (x1, y1) and (x2, y2). The formula for finding slope is



This formula is where some of the different descriptions of slope come from. Since moving from one y-value to another is an up/down motion, some people refer to this a rising. And since moving from one x-value to another is a right/left motion, some people refer to this a running. And that’s where the expression “rise over run” comes from. The formula also shows why some people refer to slope as “the change in y over the change in x”.

One thing that some students have trouble with is the notation in the formula. The 1 and 2 that are used below each x and y are just used to indicate that we are pairing up a certain x with a certain y. The notation is a way to label the first point and the second point, and the first point has an x and y coordinate ( x1 and y1) and the second point has an x and y coordinate (x2 and y2).

Think back to the original points of (-1, 3) and (4, -2). It will not matter which one you call the first point and which one you call the second point. The important thing is to label them and then use the formula to compute the slope. Let’s use those points in an example.

Last Practice:
  1. Find the slope of the line that goes through the points (-1, 3) and (4, -2).
Begin by labeling the points. When you are first doing these computations, it may be helpful to write everything out to avoid confusion. As you become more proficient at calculating slope, you may be able to eliminate some of the steps.
Using these values in the formula gives
It is not necessary to put each value in parentheses, but it is a way to ensure that signs and numbers are accurate.

You might find it interesting that if we had labeled our points differently, the answer would still be the same. By using
we would get
Are There Special Cases?
Yes, two. All the lines shown so far have either a positive or a negative slope, have equations where slope can be found, or points that make use of the slope formula.
  1. Horizontal Lines.
Consider the line y = 3.



This horizontal line does not go uphill or downhill. It is flat. What does this mean in terms of slope?

Since all y-values are equal to 3, let’s choose any two points on that line and use the slope formula. Just choose any two x-values and look at the points (2, 3) and (-5, 3). Now apply the formula.
You should see that no matter what two points are chosen, the change in y will always be 0. And anytime you have 0 in the numerator of a fraction, the value will be 0.

In general, anytime you have a horizontal written in the form y = k, the slope will be 0.
  1. Now that we’ve considered a horizontal line, what about a vertical line?
What if we have the line x = -4?



Just as we did with the horizontal line, let’s choose any two points on the line. All the x-values will be -4. Let’s use the points (-4, 3) and (-4, 5). Using these values in the slope formula gives
No matter what two points are chosen, the change in x will always be 0. And anytime you have 0 in the denominator of a fraction, the value is undefined.

In general, when you have a vertical line in the form x = h, the slope in undefined.

Be very careful about the distinction between an undefined slope and saying no slope. Saying no slope is that same as saying the slope is 0 which is NOT the same as undefined. The terminology you use is VERY important.
Knowing the slope of a line is a valuable tool when graphing a line. To learn more about graphing lines, click here.

Examples
Example What is the slope of the line 2x - 4y +5 = 0?
What is your answer?
 
Example What is the slope of the line connecting the points (6, 9) and (0, 9)?
What is your answer?
 
Example What is the slope of the line connecting the points (6, 9) and (-2, 1)?
What is your answer?
 
Example What is the slope of the line x = 6?
What is your answer?
 
Example x = 4y - 8
What is your answer?
 



S Taylor

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