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Lesson 130: Introduction to Slope and Y-Intercept
In this very important lesson we'll learn the basics of graphing linear equations.
Before starting this lesson, make sure that you review Lesson 100 which introduces the concept of linear equations. Any equation of the form y = mx + b, where m and b are particular numbers, is a linear equation. That means that if we were to make a graph of that equation, it would look like a line. It might be vertical, horizontal, or diagonal.
In this lesson, we'll learn how to look at a line on a graph, and easily determine the equation that generated it. Look at the blue line on the coordinate plane below. First let's look at where the line crosses the y-axis. We can see that it does this at -2. The coordinates of that point are (0,-2), and we call that point the y-intercept. We typically use the letter b to represent the y-intercept, and that is what the b means in the linear equation pattern above.
Now let's look at the steepness, or slope of the line. We define slope in several different ways, all of which mean the same thing. Sometimes slope is defined as rise/run ("rise over run"). That means that starting from the Y-intercept (or any point), we look to see how far up or down we have to go, and how far left or right we have to go to get to the next point on the line.
If you examine the blue line, you can see that the next point that the line clearly intersects is (1, 1). The point after that is (2, 4).
Our rise is 3. That means that to get to our next point we must go up 3 units. Our run is 1. That means that to get to our next point, we'll also have to go 1 unit to the right. Slope is rise over run, so our slope is 3 / 1. That can be simplified to 3.
Sometimes you'll hear slope defined as "change in y over change in x". That's really what we did. We saw how the y coordinate changed between two points, and divided by the way that the x coordinate changed. For example, let's look at (1, 1) and (2, 4), which are two points on the line. We can do 4 - 1 to see how much the y coordinate changes, and it's 3. Then we can do 2 - 1 to see how much the x coordinate changes, and we it's 1. That's how we get 3 /1.
Sometimes you will hear slope defined as "delta y over delta x." The Greek letter delta looks like a triangle, and we can write slope = Δy / Δx. Delta just means "change in." Yet another way of thinking about slope is "vertical change over horizontal change." Note that we typically use the letter m to represent slope, and that is what the m means in the linear equation pattern of y = mx + b.
We are actually ready to write the equation of the blue line.
Our pattern is
y = mx + b. We know our slope m is 3, and our y-intercept b is -2. We can therefore write y = 3x - 2 as our equation of the line. With this equation, we can easily find the y value for any value of x. All we do is substitute our value of x, and determine the value of y, and then plot the point that we get. You will see that it will be a point on the line.
For practice, let's find the equation of the red line. We can see that the y-intercept is 1. Our rise is 1. We have to go up 1 to get to the next point. Our run, though, will be negative. We have to go to the left, and not to the right, to get to our next point. We have to go to the left. That means that our slope is 1/-2. Usually we would write that with the negative sign in the numerator, but it works out to the exact same thing. Depending on where we put the negative sign, we either go up 1 and left 2, or we go down 1 and right 2. Think about why both of those are the same thing.
We now know our slope and our y-intercept. Our equation of the line is y = (-1/2)x + 1.
Make sure that you study this lesson hard, and fully understand it. This forms the foundation of all of the graphing work we will learn in later lessons.