 So in chapter one, when we talked about average rates of change, we talked a lot about slope because if you have some function, right, the average rate of change is trying to approximate the function using a straight line, this so-called secant line. The thing about linear functions is that their average rates of change are always constant and that constant value is the slope of the function, which the slope will compute by the usual formula. You're going to be taking your rise over run here, that as you take your change of y, the delta y over the delta x, that formula doesn't change. What change or what's special about linear functions though is that when you start choosing specific y-coordinates f of x2 and f of x1, it doesn't stink and matter which choice you make. Every choice of x is going to lead to the exact same slope. The average rate of change of a linear function is always constant. That's what makes it a straight line, after all, and that's of course the slope of the line. And so a line is going to be described by a linear equation and with the exception of vertical lines, every linear equation can be written in the form y equals mx plus b. And again, as long as we don't have a vertical line, this linear equation will give us a graph passing the vertical line test and it creates a function, which we call a linear function, right? So this equation of the line, this y equals mx plus b, is commonly referred to as the slope intercept form of the line. Like we mentioned before, if you plug in x equals 0, you're going to get y equals b. And so b right here is the y-intercept and like we've talked already, m here is giving us the slope of the line. It tells us how quickly we incline, right? If your rise is so much and your run is so much, this ratio of rise to run, the change of y with the change of x, the change of the vertical with respect to the change of the horizontal, will tell you how quickly this thing is rising or falling. I should mention that when your slope is positive, your linear function will be an increasing function. On the other hand, though, when your slope is negative, so maybe you have a negative rise, which means you're falling and then you have a positive run. I usually think the run is always being positive. Rise could be positive or negative. In that situation, if you have a negative slope, this will actually produce a decreasing linear function. And one thing I should mention about the monotonicity of a linear function because of the slope being constant, the linear function is always increasing or always decreasing. There are no switches. There are no extrema on a linear function that never switches from increasing to decreasing or vice versa. In terms of monotonicity, it'll always be increasing, always be decreasing determined by the slope. Also, in terms of concavity, a linear function will never be concave upward. It'll never be concave downward. By, you would sort of expect it, a linear function will always be straight, that neutral element in terms of concavity. So those are important things to be aware of as you're graphing linear functions. And so when you know these principles, linear functions, you draw straight lines, right? Always increasing or always decreasing, never concave up, never curving down. When you know how to draw a linear function, then it starts, we just have to ask the question, what's linear function do we graph if you have an equation like y equals mx plus v? Well, a strategy that many people will use is that if you know the slope and you know the y intercept, you plot the y intercept first at the point zero comma B and then you'll go up by the rise over by the run, you'll find another point and then you connect the dots and that gives you your linear function. So if we have the slope intercept form of the line, we can very easily graph that line. But what if we don't have an equation of the line? How do we, how do we graph it? How do we find an equation of the line if they're not given to us, if we're not given the slope intercept form? This is something that one sees a lot, that when we are, say, collecting data about the relationship between two quantities, no one's giving us the function. They're giving us the data for which there's a function relationship between the two quantities. We then have to find an equation, we have to find a function that models the data accurately. And so this idea of curve fitting, finding a function that fits the data is a very important principle. So we have to ask ourselves if we have some data about a linear function, how do we come up with the equation? Well, if you have two points on the line, so we have their x1, y1, and you have x2, y2. Turns out we can find the slope of the line by the slope formula. This is a slope. The formulas for slope is something you're going to want to know. We can find the slope given any two points on the line. Then it comes down to how do we find the Y intercept? Excuse me. The Y intercepts, well, one way of doing is once you find out the slope, you just take the equation 0 equals m. Sorry, you take the equation, you just take the equation up here and you're going to plug in some point, you know. So you'll get something like y1 equals mx1 plus B, where B is the unknown and you solve for B. So you can basically solve the slope intercept form for the intercept and then plug that point in later. We'll see an example of that in just a second. Another approach that one takes is you use the slope formula where one of the points is determined, say, a comma f of a. We know some x1, y1. But then we leave the other, the x2, y2 as variables. We have this x comma y. And then we take this equation and we solve for y. This would give us back slope intercept form. But in the interim, we get an equation that looks like the following. This is often referred to as the point slope form of a line. And it often works as a good starting point for finding equations for lines. So with the point slope form, what you have here is the slope is part of the line, the part of the equation. You're going to have a specific point in mind. You have an x1, y1, so a specific a and f of a right there. And then x and y are treated generically. And so in this situation, when you have a specific point, this gives you an equation of the line because there's not a unique equation. But then we can transform this equation into the slope intercept form. Let's look at some examples of this. Consider the following line that's given. We want a line that passes through the point three comma four. And we know the slope of the line is negative two. And then, of course, if the slope is negative two, that means you go down to one to the right and that gives you another point on the line. We can connect those dots here. Let's focus on finding the equation of the line. So because we know the slope is negative two, do make sure you keep the negative sign here. This is a decreasing linear function. We have the slope. So one way of trying to finish this up here is to take the slope intercept form, which we have now a negative two x plus B. We don't know what B is, so let's solve for it. If we substitute, if we substitute our point into the equation, because three comma four being a point on the graph means that when x is three and y is four, the equation will be balanced. So we plug in x equals three, y equals four. This would give you four equals negative two times three plus B. You could then start solving for B in this situation, right? B minus six two times three is equal to four. We're going to add six to both sides and you get that B equals 10. You would see that's the y intercept. And therefore our equation looks like y equals negative two x plus 10. So many of you will like to find the slope intercept form in that situation. Whenever you're looking for the equation of a line, just assume that you were supposed to put it in slope intercept form. So we got y equals negative two x plus B. That's if we just want to solve for the y intercept directly. An alternative approach, which is actually the one I kind of prefer, is if you want to use the slope, the point slope form, y minus four equals negative two times x minus three. So the point slope form is you put in the slope and you put in a specific point on the line, which we have three and four right here. Now you can simply just solve for y to put in slope intercept form. That requires distributing the negative two. So we get negative two x plus six. And then we have to we have y minus four. Then add four to both sides and you'll see you end up with the exact same thing. Y equals negative two x plus 10. Both processes will give you the exact same slope intercept form. It just kind of comes down to what do you want to how do you want to do it? Do you want to plug in the slope, plug in a specific point and then solve for the y intercept, or do you want to just use the point slope form? Like we see right here, plug in the slope, plug in a point and solve for y. It both will do the exact same thing. Like I said, I prefer I prefer this one because when you're done with the algebra, you have the equation in hand. This one right here, when you're done, you find the wider set. You still have to plug it in there. Not a hard task, but still something else to do nonetheless. Let's look at another example of something like this. In this example, let's find the equation of the line through the point two comma one and negative one comma negative four, the graph of which you can see over here to the left. How does one find the equation of line? Well, when it comes to the equation of line, both the slope intercept form and the point slope form require the slope. How do we find the slope? Well, if you have the graph, you could try counting things. You're going to go up by so many over by so many. If your graph is drawn to scale, that's not that's not too hard thing to do. But honestly, we're just going to use the slope formula in this context. Right. M equals y two minus y one over x two minus x one. And so then what do we do here? We're going to choose our points, which are given to us. The first point, second point, plug them in there. We'll say this is the first point and this is the second point. It doesn't matter who's first or who's who's on second or anything like that. Just be consistent. So I'm going to say this is my second point. So we're going to get negative four minus one. I took the difference of the y-coordinates. Now just go in the same order. The difference of the x-coordinates is negative one minus two like so. Combining terms here, we're going to get a negative five over a negative three. That's a double negative. We're actually going to get five thirds, which after all, this should be a positive number since our line, as we can see, is an increasing line. So we get our slope, which is five thirds. Don't panic when your slope turns out to be a fraction. That's actually going to be more common than not, because when you see this fraction, this ratio is just oh, there's a five to three ratio between rise and run. It goes up five every time it moves over three. That's all that it is here. Don't be panicked by the fraction. And then once you have your fraction here, you have your slope, we then solve for the equation here. I would start with point slope form. So we get y minus you have to pick your favorite point. I'm an optimist, so I'm going to choose these positive numbers right here. So we're going to take y minus one is equal to five thirds times x minus two. And this gives us an equation of the line. But to put in slope intercept form, distribute the five thirds. We're going to get five thirds x minus 10 thirds right here. We have to add one to both sides. But to kind of help myself out here, I'm actually going to add three thirds, which is the same thing as one, so that I have a common denominator. And then adding this together, we get y equals five thirds x minus seven thirds right here. And so our our our y intercept here is going to be negative seven thirds. We we'd see it's below the x axis, and it's going to be a little bit shy of, you know, it's going to be a little past negative one, which is perfectly fine. Fractions are our friends, not our enemies. And we did, you know, it's a little bit more tedious to do arithmetic. You know, we have to add fractions together, but we can handle it. Believe in yourself. I know you can do it. And this shows you how we can find the equation of a line if we know some information about the line, like points on the line, slope of the line. Though that's the information we want. We want the slope on the line. If we don't find if we don't know it, take your two points and use the slope formula to find the slope of the line.