Okay, great lesson here. Planned linear functions and slope. I hope a lot of this will be reviewed for you from your high school algebra, but we'll see how that goes.
Let's begin. Here are some of the things we want to look at. Be able to find slope. Be able to write the equation and graph the equation of a line.
And that's, I'll get into the details, but that's these two. Find slopes of equations. parallel and perpendicular lines, and then average rate of change. So let's go ahead and begin. Okay, for this first objective, our goal is just to find the slope of the line given two points.
So what we've got is we need a formula for slope. The letter that's used in math class is m, and what it is is y2 minus y1 over x2 minus x1. Another way to say that is the change in y over the change in x.
So when I look at these points that are given to me, I can say, well, this happens to be the first one. The order here does not matter. But I say, well, here's an x and here's a y, so these are my first ones.
And here's an x and here's a y, and these are my second ones. So you could do that and you can hit it every time. So then when you want to know the slope, you simply just take and use the formula. Take your y2, which is negative 9, and subtract your y1.
Then take your x2 and subtract your x1. Now you have to be careful. Notice these signs can change up quite a bit. So that's a negative 15 on the top.
And this becomes 2 plus 3. So it looks like we have a slope of negative 3 for that first one. Again, let's practice that with the second one. Here's an x and a y, the first ones.
Here's an x and a y, the second ones. So my slope would just be... Take y2 minus y1, so that would be 8 minus a negative 1, and then 5 minus a negative 3. So we would have a 9 on the top, an 8 on the bottom. That would be my slope. If you can reduce the fraction, this one didn't, but if you could reduce it, you'd obviously try to reduce it.
Again, let's hit it again. Here we've got an x and a y, the first ones, x and a y, the second ones. And so I've got slope is negative 6 minus a negative 6, and then a 10 minus.
You see how I'm trying to emphasize that minus is just part of the formula. It's the difference there. what's the x? 7. So you have a negative 6 plus 6, that's 0. 10 minus 7 is 3. 0 thirds is just 0. Now I've got to give you a note here, because 0 slope means that the line doesn't have any tilt to it, which means it's going to be horizontal.
It's going to be flat, like the horizon you see the sun on. It's a flat line there. Okay, back over here, we've got an x and a y. There's my first one, an x and a y.
There's my second one. And my slope would be a negative 9 minus a 2 and a 3 minus a 3. So in the top we have a negative 11. In the bottom we have a 0. Uh-oh, can't divide by 0. That's considered undefined. So that's what we would say.
It's undefined slope. Hey, I can't help but tell you here, this is going to be a vertical, vertical line. It means it runs up and down. Undefined slope.
Hey, while we're at it, this one over here, the second one was... positive slope. So that means you're going up.
As you read from left to right, you're going up. This one's a negative slope. So it goes down. How much was dependent upon what the number is, but that's kind of a picture of what's going on.
So you see negative slope. And if this is kind of new or it's been a long time, make yourself some notes here. This is negative slope.
This is positive slope. This is horizontal. You can say zero slope.
Oops, slope. And the other one. The other one down here is vertical or undefined slope. Okay, all right, our next concept is write and graph the equation of the line given its slope and one point. Okay, so we're going to need some formulas again.
So we have... There's a couple of different ways. I'm going to try to keep this very straightforward. We've got this formula. It says y minus y1 equals m times x minus x1.
Now, if you have another way of doing it, feel free to do it the way, if you learned another way in high school or something. I'm trying to pick the way that's probably the most common, and most of the books and most of the students I've been working with over the years, this is the one they usually go after. So what we do is we've got this m value.
See, that's already given for us. And this is considered an x and a y, our first one, in this case our only point. So those values would plug in right here.
So when I use that formula, I've got y minus this y1 value, so a negative 1. equals m. So I come back over here and I grab that two-fifths. And then I have x minus.
And then I look back over here and sub in a four. So you want to be very mechanical, very straightforward. Just plug those in. Now I will clean it up a little bit. And I will get y plus one.
And in here, we'll just take and distribute. So simply just send that thing through. You can write that as 4 over 1 if it helps you to get that figured out.
So we've got here 2 fifths that multiplies the x, and then this is going to be 8 fifths. My next step... is to subtract 1 from both sides so you can get the y by itself.
Well, subtracting 1, now notice here, this is just the way I do it. Okay, I want you to see something. See, I'm subtracting 1, that's going to get rid of those.
And I could write a 1 here, but I'm going to grab a common denominator because it's so much faster. So if it's fifths, I call it 5 fifths. If it's sevenths, call it 7 sevenths, you know, whatever. And then you can come over here and just call it y. equals, and here's my answer, 2 fifths x minus 13 fifths, because I already had that common denominator in place.
And I want you to remember a couple of little things here. We've got this formula is called point-slope formula. I mean, that's nothing fancy, but if you were looking this up, you want to go look it up and see, you know, someone else working or whatever. But you've got point, there's your point, the x1, the y1, and there's your slope.
So a point-slope formula. This one is called slope-intercept form because this represents... The m right here, so you've heard this before, I'm pretty sure.
y equals mx plus b. Well, that b, that b value is called your y-intercept. And of course, this thing is just called your, the m is just called your slope. So you've got a slope and a y-intercept. So my answer, is y equals two-fifths x minus thirteen-fifths.
Now, I want to put a graph in here for you. So I'm just going to put it to the side and just erase a little bit of this. Now, I'm putting a graph over here, and I'm trying to be as straight as possible.
This is what I want you to do. Just practice. These don't have to be just perfect, perfectly graphed out with graph paper.
You don't have to do that. Now, the way you graph this, this is the important part I want you to pick up on. The way you graph this is you start here at this y-intercept.
So I'm just going to make a note here. So negative 13 fifths is the same thing as negative, how many times does 5 go into 13? Two times?
That would be 10 with 3 left over. So you want to come down to minus 2 3 5ths. So when I'm looking at this on my graph, you want to get it as close as possible. Trying to give myself a little bit of room to work. So if I come down minus two and three fifths, I'd be about right there.
Then what I wanna do from that y-intercept, you see it's right on the y-axis, then I wanna rise two and run five. Remember, and if this is new to you, you probably wanna make yourself a note right here, the slope is rise over run. And you notice both of these are positive values.
So back here on my graph, I'm going to rise 2. So come over and just count it off your paper there. So there's 1. There would be 2. So I'm rising 2, and then I'm running 5. Positive direction means I'm heading over this way. So... come over one two three four five you got to stay in line with this guy and put you a point up and that's the second point now see once more if this is fairly new to you so you've got your y intercept rise two run five set your straight edge on there and connect your line it goes right through there And that would represent the graph.
Let me put little arrows on there. That represents the graph of this 2 fifths x minus 13 fifths. Now pause. Notice here that it's got this positive slope. As you're reading from left to right, it's increasing.
Okay, so that's one of them. Let's go ahead and look at this second one. Let me clean the board off. Okay, this says the slope is negative 7. and the x-intercept is 3. Now I have to visualize some of this stuff so what I'm going to do is just in my mind I'm picturing this x-intercept so I'm just going to draw myself a little note right here.
Nothing fancy but I have that picture in my mind and you're telling me that the x-intercept is 3 so that means I'm touching the x at 3. So what would be the coordinate of that point? It'd be 3, 0. Good. So that's going to represent up here in my formula, that's going to represent my x and y or x1 and y1.
So let's start to put this thing together. Using this formula, we've got y minus My y value, my y value is just 0. My slope is negative 7. x minus my x value that's given, that's just 3. So y minus 0, that's just y. Here we want to go ahead and distribute, distribute that thing through. And you get negative 7x plus 21. Now we want to graph that.
Okay. Let me get my graph paper set up. Okay, as I begin to put this graph together, I want you to look at this y equals ilmex plus b.
This is my b value, and b is my y-intercept. So my y-intercept has to be up here at 21. Well, how am I going to, I'm not going to draw. 21 little marks across here.
And I notice my slope is 7. Look, 7, 14, 21. I can count by 7s across here. I can say this is 7, this is 7, this is 7, to kind of make it look like that. And then if I'm ready to put my graph together, I might as well, you know, I...
I could go ahead and count these off just as 1, 2, 3s. And if you want to count these as 7s, you can. But when I plot this, I see that I have a y-intercept of 21. And then it says, look here, it says fall 7. Because remember, this is rise over run. It says fall 7. That's the minus sign.
So I want to fall 7, which would put me right there on the 14. and then run over one, which would put me about right there. And then see, you can fall seven and run one. Oops. You can fall seven and run one, and you just keep following along that line.
I connect those dots, draw myself a line through there, and that would be the graph. You notice it's got a negative slope. Intercept negative slope means it's decreasing as we go along.
Very well done. Okay, well the next level is very similar. It says write and graph the equation given two points. Two points.
Now my common sense would be just, hey, graph those. Graph those, plot those two points, connect the dots, and move on. I think that's what I'll do first, just to play with it. So let me get my graph set up. Okay, so I've got negative 4, 4, I think, and 2, negative 8. I'm counting by twos.
Let me just count by twos all the way through here. So that would be 2, 4, 6, 8, and so on. Now, when you start doing this, you need to indicate with just two or three points is fine.
Whoever's reading your graph, me in this case, I'll be able to, if you give me two or three, I can count it. So it'll be a lot easier. Okay, so we have negative 4, 4. So that'd be 2, 4. So negative 4 in the x, and then 4 up.
So there's one point. And then 2, 2, 4, 6, 8. These are 8 down. So let me try to draw the line through there.
Pretty close. We'll find out how accurate my drawing looks in just a couple of minutes. So now I've got the graph.
Yay. Now I want to go and find the equation. So keep in mind, we've got this formula that says y minus y1 equals m times x minus x1. But see, You say, well, I've got two points. You only need one.
You say, yeah, but they didn't give me an M. That's your clue. You're going to have to use this formula also, which says you've got to put both of these together.
So in this first example, let's go find the slope first. So maybe I should label these points. Here's an x, here's a y.
Those were the first ones. Here's an x, here's a y. And it doesn't matter. You can reverse those if you want. They're still on the same line.
So we're looking at y2, so negative 8 minus a 4, and a 2 minus, and that's a negative 4. So that's a negative 12 on the top. 2 plus 4 is 6. So I have a slope of negative 2 when I reduce that. So the next step is to use the formula this. this top formula up here. I'm using both of these.
First one, in case you want to see a recipe, this is the first one. This is the second one that I'm going to use. Okay, so we've got y minus 4. I'll use that point.
Negative 2 times x minus my negative 4. Watch those negatives now. Now you could switch. You don't have to use this.
I know I've labeled it x1, y1, but you could use x2 and y2. You'll get the same answer. Okay, so that's nice. Very flexible.
Okay, what do we have? We have a plus right here. Let me go ahead and distribute that negative 2, and then I'll add the 4. And that will give me negative 2x, and that's what, minus 4. We'll see how close my squiggly graph over here got.
Yeah, wasn't very, very perfect. Let me see if I can clean my graph up. Because, see, you say, how do you know it's not very perfect? Well, I should be going through at negative 4. You see how I'm off a little bit?
But that's me just messing around with it. lines. Let me see if I can show you something here. See, these are just not super accurately measured.
See if I can get that to line up. If I tilt my ruler just ever so slightly, it's a little bit crooked, but best I can free hand on here with the ruler laying across my screen. I am looking at Pretty close to my minus 4 right there.
So you can see that they all are in agreement, and that's what I really want to focus on. Let's try one more, just for practice here, make sure we've got it. Okay, so for these two points, we're going to use the same kind of idea.
We're going to need that slope formula. Let me just write them again. Very first thing, let's go find that slope.
So that means I need an x and a y. These are my first ones. These are my second ones.
So 7 minus a negative 1 and 4 minus 4. Now we see something weird happening here. See that undefined? undefined.
Let's look at a picture of the graph. So I've got the point 4 and negative 1. These are just counting by ones here. There's the point 4 and negative 1. And then we have the point 4, 7. 2, 3, 4, 5, 6, 7. So somewhere up here.
And you can see when we connect the dots there, what we have is that vertical line running right through those two points. Vertical line, that's what this is talking about. Undefined means we're going to end up with a vertical line. If you didn't know that, you need to write that down.
undefined slope. So you say, what's the equation? You know, because when I'm trying to work over here and I've got my next formula, you say, what do I put in?
What do I put in for that slope? You know, because that slope just erased. What do I put in for that slope right there? If that's undefined.
then you don't need this formula. You just come over here and you call it. Now, this is where the common sense is going to kick in.
What's my x value here? 4. What's my x value here? 4. What's my x value for any point on here? 4. So the equation is x equals 4. So we wrote the equation. And we graphed it.
So listen, vertical line. So here's some notes for you. Vertical lines are always of the form x equals and then some number, whatever the number is that you go through. Hey, and I didn't give you an example of this one, but if you want to play with a horizontal line, and you can actually use the formula if you want to because you have zero slope.
We're talking about zero slope. Well, these are always of the form y equals and something through there. So anyway, just a few more notes on that.
Okay, let's keep playing with this concept of slope. There are some of the basics. Okay, we're looking at parallel and perpendicular lines now. Parallel, here are your notes.
They're pretty short. I can fit them on this same screen here. Parallel means we have the same slope.
I mean, visualize a couple of lines going through here. Here's one line, and here's one line that's running parallel. Well, they have the same slope. You say, what about perpendicular? Perpendicular, that's a little more complicated.
perpendicular means you have and you can say this different ways but i call it opposite reciprocal opposite reciprocal slopes the picture would be something like you've got a line going through here and then you've got another one that's crossing through here and these are at a right So they're intersecting at a right angle, but you have to use the idea that they're opposite reciprocal. So I'll practice that with you. Okay, so my first example here says find the equation of the line that passes through this particular point and that is parallel.
So where you want to start, you know, step one is this part right here. Because there's really only two steps here. Step one is find your slope.
Well, if it's parallel to this thing, so what is this equation? Well, let me clean it up here. What if we move the 2y over, and what if I bring the 7 back to this side?
I'm just trying to keep things positive. If I divide everything by... 2. That means the 3 gets divided by 2 and the 7 gets divided by 2. This is y equals l max plus b.
So my whole reason for doing this little step right here is to get that slope. So parallel means what? Same slope. So we know For this particular exercise, we know that we want our slope to be the same as this one, so that would be three halves.
And then this is just some given x and y. We'll call it the first ones. And now we can use that formula that we've been using over and over and over, called the point-slope formula.
It's just simply a tool to help us build a line. So we've got y minus my negative 5 equals, we're keeping that same slope, 3 halves, x minus x value. So that reads y plus 5 equals, here we're going to distribute.
Now if it helps you to write that over 1. So when you distribute through here, you can just multiply straight across, then do that. So we have 3 halves times x. Positive times negative is negative.
That's 6 over 2. 6 divided by 2 is just 3. So now we can subtract 5 from both sides, get this thing to where you get the y by itself. 3 halves y equals 3 halves x. minus 8. And that's how you play the game. So step one is find the slope. And then step two is just plug it into this formula.
Parallel means same slope. Perpendicular, opposite reciprocal. Let's try.
I gave you two examples here. Let's try the second one. So in the second one, we're looking at the end. equation of the line that passes through the point perpendicular. So the key word here is perpendicular.
So the first thing I want to do is take this equation, solve it for y. In this case, I think it'd be easier just to move the 4x over. And all of a sudden you're in slope intercept form. y equals mx plus b.
So this is my slope. So perpendicular up here means opposite reciprocal slope. So the slope that I need to build the equation of this line is the opposite reciprocal.
Well, opposite is going to be a positive 4, and reciprocal of 4 over 1 is 1 over 4. Here's my given x and y, x1 and y1. Go back to my formula, point-slope formula, and we'd be in good shape. y minus 7, and then 1 fourth, x minus a negative 6. Yeah, we're in really good shape.
So... We make this a plus 6, so that'd be x plus 6. And again, I'm just going to distribute through there. If it helps you because we're playing with fractions, you can write that over. But we're looking at y minus 7 equals 1 fourth x.
And then that's going to be plus 6 over 4. We really need to reduce that 6 fourths. Let's do that. Let's call that 3 over 2. And now we're going to add 7 to both sides. Add 7. Wait a minute. Hold on.
7. How many halves would that be? Multiply 7 by 2. It would be 14 halves. There we go. 14 halves is the same as 7. So we're adding 7 to both sides.
equals one-fourth x. So that would be 17 over 2. That's my y-intercept. There's the equation of my line. It's got opposite reciprocal slope.