Math 1 - 1.6.1 Notes
Math
Learning Linear functions
Okay, guys, we are jumping right into two 6 one-intro notes. This is our next unit. And we're really going to focus on linear right now. So identifying the Y-intercept and rate of change, which is also known as the slope for each of these functions. So starting with Y-intercept, our Y intercept is when we intercept with our Y axis when we cross it. It is also when X is equal to zero, and remember you want to always write your Y intercept as an ordered pair, where X again is zero, and then you have some Y value. Rate of change, which is our slope. That's going to be our rise over run.
So just a quick little bit of review there. So first we have two equations starting with number one. This is written in slope-intercept form. So just by looking at it, I can identify what my slope is. Remember, Y equals X plus B, where M is going to be what is on my Y intercept or my rate of change. M is my slope, which is my rate of change. So that's one over four, easy peasy. And then B is my Y-intercept, so my Y intercept here would be negative 5, but I want to make sure I write it as an ordered pair. So zero, negative 5. Now sometimes when you have equations, it's not always going to be given to you in the form Y equals MX plus B so number two, I need to get it into that form. I want to get Y by itself. So I'm going to subtract two X on both sides to do that. And I get Y equals, and then I'm going to start with my X sometimes I'm used to seeing it. Plus three. So what would my winners have to be here? Remember it's that B value so that would be zero three, and then my slope would just be negative two. Okay, so hopefully the equations are not too bad.
The next type of former and a C here is graphs. So with graphs, I want to identify my Y-intercept and my rate of change. Remember, Y-intercept is when I cross or intercept my Y-axis. So I'm looking to see where that line crosses, and it's going to be right there at that point. And I don't want to just say three, I want to write it as an ordered pair, so that would be zero, three. And then with my rate of change, I need two XY coordinates where X and Y are both whole numbers. So I don't want to pick a weird spot like I picked here, right? That's not my line, but could you tell me what my Y is? Not really. It kind of looks like 1.5, but we're not sure. So we're not going to use that point. Instead, I could use this point where it's really easy to see okay? My X is two and my Y is zero.
From here, I need to see, okay, how much am I going? How much am I rising and how much am I running? So then I count out. Okay, one, two, three. My rises three, one, two, and my run is two. Swimmer, it's rise over run, so that would be three over two. Last thing is really pay attention to the direction of your line. So my line is going downward, and that's going to tell me that my slope here is going to be negative. So I want to make sure I include that. So my slope or rate of change would be negative three over two. Okay, for the next problem, my Y-intercept. Go ahead and take a moment. What is my Y intercept going to be here? Make sure you write it as an ordered pair. Check to see okay, where do I cross on my Y axis? Bam. Right there. At zero, negative four. Okay, my rate of change, I'm going to pick a point again, and I want my X and Y to be whole numbers.
So I'm going to pick right here. And then I count. One, two, three, so my rises three, one, one. So my rate of change would be three over one, which is just three. Now, if you picked a different point than me, that is okay. So let me. Show you. So let's say I did not pick that point. Still use the Y on our set. But maybe I saw this point up here. I would still count one, two, three, four, 5, 6, so my rise is 6, and I ran one, two. So remember rise over run 6 over two, which sticks over two is just three. So even if you don't pick the same point as me, you still get the same answer. So don't overthink which point you pick, just make sure again X and Y are whole numbers. Okay, last up we have tables. So for 5, I want to identify my Y-intercept. Again, Y-intercept is one X is equal to zero. So I want to look at my table and see where is X equal to zero. Well, right there, that is going to be my intercept.
So zero negative four. So always check sometimes on the table it's given to you. So that's always really nice. The next thing is our rate of change, so for a table to find your rate of change, it's going to be my change in Y over. My change in X and what I mean by that is I want to look at my Y and I want to see how much is it going up by down by from each number. So I have negative four and I'm going to negative two. So I need to think, did I add? Did I subtract? And by how much? So take a moment to think about that. So for this one, negative four negative two, I add, I'm adding two, then I go from negative two to zero.
Same thing, add two. From zero to two, added two. So then if I think about my rate of change, my change in Y is going to be plus two. I do the same thing for my X, so I start at zero, and then I go to one. So I'm just adding one there. And then I go from one to two. Same thing I added one. And then two to three, add one. So that would be two over one. So then my rate of change would just be two. In number 6, it's not always going to have it where our Y intercept is on the table. Sometimes we have to do it a little bit of extra work to find that. But why don't we do our rate of change first? So we can do that without any extra work. So I've got to remember I'm doing my change in Y over my change in X. So for my Y, I start at negative three, and then I go to negative 5. So I added two. Okay, cool. From negative 5 to 7, same thing, add it to, and add it to. So for my change in Y, that's just going to be two.
Now I got to look at my change in X so I went from three to four. So that's plus one. From four to 5 plus one. And from 5 to 6, so it's plus one. Okay, so it looks like my slope here is also going to be two. And then my Y-intercept. So like I said, we don't have zero on this table. And so what we need to do is we need to work backwards. So I'm trying to get X to be zero. So if I go backwards, I would minus one. So I do the opposite of what I go, what I've been doing going down. So same thing with my Y, I was adding two before now I'm going to subtract two. So that would give me for my X, that's going to be two because three minus one is two. And then I have a negative three and a minus sign two. That would be negative 5. Okay, still not at zero, though. So again, I've got to do it one more time.
So minus one, that gives me one, and then for my Y, I sold out a minus two. So negative 5 minus two, it's going to be negative 7. And then one more time to get me to zero, so minus one, there I go. I'm at zero. And then over here, minus two. So 7, 8, 9. So I've been negative 9. So there is where X is zero. My Y intercept would be zero negative 9. Okay? The next page is going to be a U try. So go ahead and pause the video. There are three problems here, one with an equation, one with a graph and one with a table. I want you to give those a try on your own. After you do, I will run through it pretty quickly. Okay, hopefully, those went well. Start with number one, identifying our Y-intercept and our rate of change. I've got to get Y by itself. So that's my first goal. So I have four Y equals negative three X plus 12. Divide by the four to get Y by itself. I'm dividing everything by four. I get Y equals negative three over four. X plus three. So my Y intercept is my B value.
So zero three, my rate of change, negative three over four. Okay, for my graph, again, Y-intercept, seeing where I intercept with my Y axis, which would be right there at that point, which would be zero three. And then my rate of change actually also gave me another point, so that would be one, two, three, one, two. So this would be three over two. And then last one, the table. So again, let's look at my change in Y first. So 7, 8, 9, I'm adding two. And then yep, it looks pretty good. Everything looks like it's being added by two. Okay, then on my X looks like I'm also adding by two because I got from two to four. So then rate of change, I've got two over two. So that's going to be one. So my rate of change here or my slope is just one, but then I need to work backwards because I don't have zero on my table.
My X value, I don't have it as zero. So what I can do is I added two, so I can subtract two. Oh, I picked the wrong color. So subtract two. Just go up and go backwards by the same amount. So two minus two is zero. And then I can also go boop here. So that would be minus two. 7, 6 5, so it'd be 5. So my Y intercept is going to be zero. 5. And maybe you didn't go up or backwards by two. Maybe you use your rate of change. So I know my rate of change here is one. You just have to be careful because that's when we move our X value by one. So I would have to subtract by one, and then subtract by one. So this would be one, 6, and then I have to do one more time, so minus one to get rid of the zero. And then minus one to get me to 5. So both are totally fine. Sometimes it's easier to just go by the rate of change, so you don't make any mistakes. But there you go. Hopefully that was okay. If you have any questions like always, you can always reach out for some extra help. But hopefully that was helpful. I'll see you guys later. Bye.