Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

In the previous section, we used intuition to find solutions to linear equations in two variables. It is not always the case that we can easily find solutions by inspection. So we are going to develop a more systematic approach to graphing that does not rely on intuition. To do this, we’re going to build off of the solutions of lines that we looked at in the previous section.

You might have noticed that when we had charts of values that you can find other charts by changing the \(x\) and \(y\) values in a consistent manner. This sets up the idea that points on a line satisfy a certain ratio with regards to how the coordinates change. This can be formalized as the concept of the slope of a line.

There is a logical reason for not trying to apply this to vertical lines. In a vertical line, there is no change in the \(y\)-coordinate which means that we would be dividing by zero, and that leads to an undefined expression.

If the slope is positive, the line points from the lower-left to upper-right, and if the slope is negative, the line points from upper-left to lower-right. If the line is horizontal, then the slope is 0.

There is a formula for the slope of the line if you have the coordinates of two points. The formula is commonly given to students, but it can sometimes be a distraction. If you are able to construct a meaningful picture of the line, you will not need to memorize this formula. There are some practical uses from an algebraic perspective, but it’s far better to have an understanding of the concepts than simply trying to apply formulas blindly.

The slope is one of the two parameters in the slope-intercept form of a line. The other parameter is the of the \(y\)-intercept. The \(y\)-intercept is the point where the line crosses the When this point is not a point on the grid, students often try to estimate the value. This is somewhat acceptable when sketching a graph, but it is usually not acceptable when reading a graph. The challenge is that is can be very difficult to identify them correctly.

Try to estimate the of the in the two graphs below, and you’ll understand why estimation is not a reliable technique.

This highlights the importance of developing an algebraic method for working with lines so that we can avoid needing to make estimates. Specifically, if we’re given the equation of a line, it is a common goal to write it in slope-intercept form.

The reason this works is because the of a line is the solution when \(x = 0\text{.}\) But in this form, setting \(x = 0\) makes the first term disappear, and you’re left with \(y = b\text{,}\) which tells us that the line must pass through the point \((x,y) = (0,b)\text{.}\)

Once you have the equation written in slope-intercept form, you have the information to sketch the graph. You can immediately identify the and then you can use the slope to find a second point.

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Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.

There is a lot more than can be said about the use of lines in applications, but those will have to wait for their corresponding courses. At this point, the goal is that you are comfortable enough with the algebra of two-variable equations to convert them into slope-intercept form, and that you are able to sketch the graphs of such lines.

It turns out that the idea of slope is extremely important in both practical and theoretical mathematical thinking. Slopes can be used to represent important relationships between variables, and they one of the core concepts that we see in calculus. So we want to take extra time to really think about how we can understand and interpret the notion of slope.

Let’s imagine that we’re watching someone run the 100-meter dash, and that it takes them exactly 10 seconds to reach the finish line. It would make perfect sense for us to say that they ran 10 meters per second. But what does that really mean? Let’s look at a graph that represents the situation.

If the original fraction has a familiar feel, that’s because it’s the formula for slope. It’s nothing more than the rise over the run. We can see this more clearly if we connect the two points to create a line.

But there’s a problem with this graph. It does not represent the actual position of the runner at every moment in time. Sprinters don’t run at a constant speed. When they start they’re moving at a slow speed, and then speed up as they go. So the real graph may look more like this:

So let’s go back and think about what we mean when we say that they ran 10 meters per second. This actually represents the average speed over the course of the race. If we only knew the starting point and the ending point, then we can calculate this value as our best estimate of the speed, even though something different may have happened in between.

From this picture, we can try to break things down further. What was the average speed in the first 5 seconds and the last 5 seconds? Now instead of using just the endpoints, we’re adding a point in the middle.

Notice that we now have two slopes. One slope for the first part and another slope for the second part. In other words, we now have an average speed for the first half of the race and the second half of the race. The smaller slope at the beginning and larger slope at the end matches with our intuition that the sprinter is moving faster at the end of the race than the beginning.

We can see that theses lines do a better job of capturing the information of the graph, but it’s still not very good. We can still see some rather large deviations between the straight lines and the curve. But why stop at just two divisions? Why not make a new division every second?

Now the dashed line matches very closely with the curve. It’s still not perfect, but it’s getting a lot better. We now have the average speed for every second of the race.

This leads us to an interesting question: What happens if we continue this process? Instead of every second, what if we pushed this to every half second? Or every tenth of a second? Or every hundredth of a second? We can get average speeds for smaller and smaller time intervals. How far can we push this idea? And what is the end result?