Graphing Equations Using Two Points

A generalized way to represent certain data relationships is by creating a graph of that data. A graph is a representation of all x and y values that can make the equation true. Let’s take a look at an example:

y=x+2

There are many (x,y) combinations that make this equation true. For example, if we want to see what y is when x=1, we simply substitute 1 in for x in the equation:

y=1+2

And then solve for y:

y=3

The combination of x=1, y=3 is a point on the equation y=x+2. The notation for writing coordinates or points is (1,3).

Let’s plot this point on a graph:

We now have one point on this line. Let’s find another point on the line by choosing another x (let’s try x=-1). Just as before, we’ll substitute in x=-1 into our equation:

y=-1+2

y=1

Now we have calculated the point (-1, 1), let’s plot this as well:

Since we have two points on the line we can complete our graph by connecting the points together and creating the line:

Graphing a Line Using Slope and Y-Intercept

We’ve learned how to graph a line by finding two points, now let’s try a different method. First, we’ll discuss the idea of slope. Slope is the rate at which the line rises or falls. It is also known as:

Slope=Rise / Run

Rise is the amount the y direction changes and run is the amount the x direction changes. Let’s take a look at our example from above:

y=x+2

In the previous example, we found two points on the line. We can now use these values to figure out what the slope of the line is. As we said before, slope is the rise/run (or change in y / change in x).

Our two points from the previous example were (1,3) and (-1, 1), so if we compute the change in y and divide it by the change in x for these two points, we’ll have the slope.

First, look at our graph and determine how much the graph rises from point (1,3) to point (-1,1).

The graph has a rise of 2. Now look at how much the graph runs (change in x). Going from -1 to 1 on the x axis is a run of 2.

Now that we know the rise is 2 and run is 2 we can find the slope:

Slope=rise/ run
Slope=2/2
Slope=1

That’s it! The slope of the line y=x+2 is 1.

Now let’s discuss the concept of y-intercept. The y-intercept is the value of y when x=0, or the location where the line intercepts the y-axis. To determine the y-intercept graphically, look again at the graph of our line and find the point where the line intersects the y-axis:

As you can see from the graph, the y-intercept is 2 because it crosses the x axis at y=2 (i.e. where x=0, y=2).

Slope Y-Intercept Form

So now we know about slope and y-intercept. Instead of writing out slope and y-intercept all the time, there is a standard variable that we use to describe each (m=slope, b=y-intercept). This leads us to the slope y-intercept form for the equation of a line. Once you know the slope (m) and the y-intercept (b), you can derive the equation of the line very easily by using this formula:

y=mx+b

So for the equation y=x+2, we found that the slope was 1 and the y-intercept was 2. Let’s plug the values of m and b into the formula and see what we get:

y =1x+2

y=x+2

It matches! So anytime we have a slope and y-intercept, we can derive the equation of the line by substituting m and b into the formula.