The slope-intercept form of a line is not the standard form. It is the form that I find most useful. It is also the form that I find most intuitive. Therefore, this is the form I would like to start with.
In unit 01, most of the equations we wrote, we started out writing in slope-intercept form. Consider the following problem. Cyrus is tending hod. He needs to make 80 lbs of mud for every 50 bricks to be laid. When he gets to work, one of the masons has already mixed up 240 lbs of mud. He needs to make sure that there are enough supplies. He can find out the number of bricks they expect to lay, usually between 350 and 550 bricks. How many pounds of mud will Cyrus need to mix today?
Let the number of pounds of mud be c for cement (in case you didn't know what this problem is about), and the number of bricks be b for bricks. The problem can be solved with the equation
This equation is in slope-intercept form. We should graph this so you can see what this means. We will do this in the same way as we did before: make a data table, plot the points, connect the points. We will make a smaller data table. We know this will be a line, so we really don't need too many points. This is a continuous problem, as bricks can, and will, be sliced into appropriate sized pieces as needed.
First we wish to make a data table. The domain is 350 to 550 bricks, and we only need a couple data points, so how about 350 bricks, 450 bricks and 550 bricks.
|Cement Needed vs. Number of Brick|
|Number of Bricks||Cement Needed,|
|350 brick||320 lb|
|450 brick||480 lb|
|550 brick||640 lb|
Next we need to graph this. The coordinate system we normally use to make graphs is called the Cartesian coordinate system. This is named in honor of Rene Descartes, although he never actually used it. We will learn more about Rene Descartes later in the course.
Now we can make a few observations about the graph. Take a second and make as many observations about the graph as you can. Consider things like the behavior of the graph in general, and the value of the graph at specific points. Also, take a second to articulate what your observations about the graph say about the problem. Seriously. We will discuss this below.
One thing you may notice is that the value of the function is not always positive. The line crosses the x-axis at 150 bricks; it is positive for values greater than 150 bricks and negative for values less than 150 bricks. This point is called the x-intercept. Based on the problem, what does it mean for the function to be negative? What significance does this point have in the problem?
For this problem, if the masons are planning on laying less than 150 bricks (a highly unlikely scenario) then there will be mud left over before Cyrus even starts work. This is a waste of mud, and should be avoided when possible. The significance of the x-intercept in this problem is that if the masons are planning on laying more than 150 bricks today, Cyrus will need to make more mud.
You may have noticed that the line crosses the y-axis at -240 lbs. This point is called the y-intercept. You may recognize this number from the equation we used to model this problem. This was the amount of mud that was already mixed when Cyrus got to work.
Another thing you may observe is that the graph is increasing as you go from left to right. The graph is increasing for all points in the domain. Think about the problem, and what this means in context of the problem.
Is it clear that this means that the more bricks the masons lay, the more mud Cyrus will need to mix up? That is logical. Can you think of a situation where the graph would decrease from left to right? What about a situation where the graph has a constant value?
If the graph is increasing, then the slope of the line is positive. Let's take a second and calculate the slope. The slope of the line is the rate of change and can be calculated by taking the rise (how far up) and dividing this by the the run (how far across) between two points. From both the data table and the graph, we know that the line goes up from 320 lbs to 640 lbs. This is a total rise of 320 lbs. Also, from both the data table and the graph, we know that the line goes across from 350 bricks to 550 bricks. This is a total run of 200 bricks. Therefore, the rise over the run, aka “slope," of this graph is
This reduces to
Notice that this is a positive number.
Compare our slope to equation 9,
The coefficient of b is which reduces to . This is the same value as the slope of the line.
The other number in the equation is -240 lbs. We already observed that this is the y-intercept of the graph. Therefore, for our equation, the coefficient of the independent variable is the slope, and the constant term is the y-intercept.
This is the called the slope-intercept form for the equation of a line. In general this is written
y = m x + b (eq. 11)
where m is the slope of the line, and b is the y-intercept.