Lesson #13: Point-Slope Form & Slope Intercept Form
I will be going over how to write point-slope formula equations and slope intercept formula equations.
Welcome back, my students. I will be continuing our math lessons together, expanding on the previous topic of arithmetic and geometric sequences. However, this time, they will be more commonly referred to as linear and exponential functions respectively. The two formulas I will be teaching you all are ways to find the basics of a sequence. In the last lesson, we covered finding the initial value and common difference/common ratio from a singular sequence. Because we are dealing with functions now, we need two sequences, one of them allowing us to find the initial value itself, or in this case, the y-intercept. You’ll find out more about that when we start the lesson.
I mentioned the slope intercept formula in the previous lesson when talking about explicit geometric equations. Today, we’ll be revisiting that and expanding upon it. I must specify these formulas are for linear functions. Exponential functions are written under a different formula and will be explained in another lesson.Â
Slope intercept is quite easy to understand. The formula goes as such: y = mx + b. You should already know that the y of the equation is the output. The m is the slope, and the b is the y-intercept, or where the line on the graph crosses over the y-axis. This much should not be new to any of you reading. When analyzing a graph displaying a linear function, finding the y-intercept is not difficult. Where the real work lies is in calculating the slope.
I have provided an example of a graph we can use to find the slope for. We have a linear function with its y-intercept being at 2 on the y-axis. To figure out what the slope is, we can use the rise-over-run method. This means we must find two points on the line where they perfectly cross over the grid for simplicity’s sake. We’ll be able to count how many units the points run across the grid and how many units the points rise up the grid until they intersect.Â
After counting, the points rise up by 6 and run across by 5. Now we can put the rise over the run in a fraction, 6/5. In some cases, you would need to simplify the fraction, but since 6/5 is in its simplest form, there is no need for that. You can account for the fact that it is an improper fraction and convert it into a mixed fraction or a decimal, but it is also fine to leave it as it is. Now that we have the slope and the y-intercept, we can write the slope intercept equation.
Before we move onto point slope form, there is another way to find the slope when you are only given two points instead of a table or a graph. The equation goes as such:
y2 - y1
———
x2 - x1
I apologize for the lack of better formatting. Tumblr truly is difficult. Anyways, let’s use an example of (4, 2) and (7, 3). The second point given is what is used first in the equation. The first point given is what is used second in the equation.
The answer will come out as 1/3. We can now move onto point slope form.Â
The equation for point slope form goes as follows: y - y1 = m(x - x1). This formula does not include the y-intercept, and is used when one or two points from the graph are given. When one point is given, it is usually with the slope. It is only when two points are given that you would need to find the slope yourself using the method I have just explained. Let’s say that the given point is (3, -2) and the slope is 4. When we plug in the values, the formula would go as:Â
If you are confused as to how the -2 became positive, it’s because that side of the equation initially turns out as y - (-2). The subtraction sign and the negative number cancel each other out, which makes the negative number positive and turns the equation from subtraction to addition. We are not done yet, however — we still need the y-intercept, so we’ll have to change the point slope equation to a slope intercept equation.Â
As you can see, the goal is to isolate y from the other side of the equation. First, we need to distribute the 4 outside of the parentheses to the values inside the parentheses. Since we still have a 2 being added to y, we need to subtract it from both sides. This leaves us with the slope intercept equation of y = 4x - 14.
The example above can be applied to any point slope equation, just with different values in place of the variables, of course.Â
This lesson is rather short in comparison to the previous one. I’d like to keep linear functions and exponential functions in separate lessons, so as a result, there is not much to cover in one lesson. There won’t be any homework for you all like last time; I was just curious about how interactive I could make a lesson be for my students. Have a good day.