2.6.2 Definition of the Slope of a Line
In the equation f(x)=mx, we call m the slope of the line. The formula for the slope is:
where Δy is the change in y and Δx is the change in x.
Now that we understand what m does, we'll give m a more formal name.
m is the slope of the line
In the last section, we saw that when m>0 our y values get larger as we move from left to right on our graph. Another way of saying this is that our function is increasing when m>0. Analogously, when m<0, our y values get smaller as we move from left to right. Another way of saying this is that our function is decreasing when m<0. Finally, if m=0 our y values don't change. We describe this function as being constant.
Given that we have a function that looks like f(x)=mx, we now see that we have all of the information that we need to get a general sense of which way the line points. So, for example, the next time someone asks you to sketch a function as f(x)=3x, without even plugging in a point, you can picture the general shape in your mind. This is the power of abstraction!
Let's now ask the question in reverse. Can we figure out if we're given information about the graph of the line? As an example, let's try to determine the slope of the line that connects the two points (1,4) and (6,24). Remember that the slope of the line tells us how much the y values change per each unit of x. Understanding this concept, tells us exactly how to compute the slope if given two points.
If the y values change by 20 (24 - 4) and the x values change by 5 (6 - 1), then the slope of the line is then 20/5: in other words, the amount that y changes per unit x. In general, then, the slope of the line, can be computed from two points by calculating how much the y values change divided by how much the x values change. We can write this as:
m=\frac{\Delta y}{\Delta x}
where the Δ is the Greek letter Delta, meaning "change in".
Explore!
Find the slope of the line containing the points (2,5) and (4,4). Slope =
Now, connect the two points and graph the line, then compare your answer to the graph below.
Find the slope of the line containing the points (1,1) and (8,15). Slope =
Now connect the two points and graph the line, and then compare your answer to
the graph below.
.
We've now studied a several different lines and all of them had slopes. What's the next natural question that we should then ask?
