Alright, so this is section 1.4. It's all about lines. So these are straight lines, and so what we're going to do is we're going to graph lines.
You're also going to be able to determine the slope of the line and apply slope-intercept form of a line. and you will also be able to compute the average rate of change of a function. So not only a line but a function as well.
Alright, so let's start with slope. Slope of a line. So the slope of a line is just y... y2 minus y1 over x2 minus x1, where the points x1, y1 and x2, y2 are on that line.
So let's just look at this example right here. Number one, it says sketch the line through the pair of points 4, 2 and negative 3, 5. So first, let's plot these two points. So this is our 4, 2, so I can just call it P, or I could call it 4, 2. Alright, and this is our negative three five.
I can call it Q if I want, or I could call it negative three five. Gonna connect those. Alright, and so we know lines continue on forever. Alright, so this is the line we're talking about. So we sketched the line through those two points.
And now we just need to find the slope of those two points. So we're going to use this formula for slope. And we can do whatever order we want. I'm going to let P be our first point.
So this is written as x1, y1, where x1 is 4, y1 is 2. And so Q is our second point. So x2, y2. So now if you want to find the slope of our line, we usually call slope or use the letter m to represent our slope. So that's going to be our second y value minus the first one over the second x value minus the first one.
And then we can simplify, three over negative seven, so we find that our slope is negative three over seven. That is the slope of this line that goes through those two points. Alright, so we also want to be able to talk about an equation of a line. So we have two main forms that we talk about.
You'll see a third one on here as well, and I'll point it out. The first one is the point-slope equation for a line. So you only need one point. but you also need the slope.
That's why it's called point slope. So the first thing you need is the point, second is the slope, and this is what the equation looks like. Alright, so let's just go through step by step.
So whenever you need point slope, first figure out what's the point we need. So we want to determine a point slope equation for the line through these two points, which we already found. All right, so which point? Well, you can use either one. I'm going to start with this one.
We'll do it both ways. So the point we're going to use is a 4, 2. We also need to have a slope to make the point slope formula. We found it up here.
It's negative 3 over 7. And then finally, we can implement our point slope equation. Okay, so let's look. It's y minus our y value from our point.
So y minus our y value, our y value is 2, equals the slope times x minus our x value from our point. Alright, and that's the point-slope form. of our line. So the y values go together and the x values go together.
Okay, so I could have done that, or, I'll write that a little bigger, or I could have used the other point. Alright, so I could have started with q, which was Negative 35. Our slope is the same. And so when we write down our point-slope equation, we're going to do the same thing.
y minus the y value from our point equals our slope. x minus the x value from our point. And so here it's negative, so make sure you note that. Alright, and we can simplify this to get rid of that double negative and make it look a little better.
So y minus 5 equals negative 3 over 7x plus 3. Alright, so both of these answers are correct. They both represent the same line. Okay, and so we'll find we can simplify these to point, sorry, to slope intercept form, and then you'll actually get the same answer. Alright, so let's talk about slope intercept equation for a line.
All you need is a slope and... the y-intercept, so that's where our graph crosses the y-axis. Alright, so let's do number three.
Determine the slope-intercept equation for the line through this point, 3, negative 2, which has slope-intercept. negative 4. Alright so one thing we notice is that we have a point and we were also given a slope. Alright so that should tell us we can use what kind of form. We can use the point slope equation because we have both of those things.
Alright so we have a point, we have a slope, let's just go ahead and try point slope and we'll actually do this problem multiple ways. Alright, so point slope says we can do y minus our y value equals our slope times x minus our x value. Alright, and then I'm just going to simplify this and solve for y, and that will give me my slope intercept equation. So this is y plus 4. I'm going to go ahead and distribute this negative 4 over here. Oh, sorry, this is not a 4. Whoops.
This is a 2. y plus 2. Alright, so now I can subtract the 2 from both sides. And so we're left with y equals negative 4x plus 10. And that is the slope intercept form of our equation. Alright, so you could also go straight from this form. It's a little bit more work, but you could do the same thing. Or, we'll get the same answer.
We could use... y equals mx plus b where x and y We're going to use this to figure out what our equation is so our y value is negative 2 our slope is negative 4 our X is 3 and we don't know what our y intercept is so we have to solve for that and Then put it back in this form Alright, so I'm just going to multiply here. So we got negative 12 plus b.
Add the 12 to both sides. 10 is our intercept, and that's exactly what we found here. So to answer the question, the slope intercept equation for this line is y equals negative 4x plus 10. So either way is perfectly fine.