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So what's our highest degree term here? I like to write it in that order. We have only one x squared term, second degree term. We only have one of those, so let me write it over here. Negative 3x squared. And then what do we have in terms of first degree terms? Just an x, x to the first power. | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
We only have one of those, so let me write it over here. Negative 3x squared. And then what do we have in terms of first degree terms? Just an x, x to the first power. We have a 16x, and then from that we're going to subtract an x. Subtract 1x. So 16x minus 1x is 15x. If you have 16 of something and you subtract one of... | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
Just an x, x to the first power. We have a 16x, and then from that we're going to subtract an x. Subtract 1x. So 16x minus 1x is 15x. If you have 16 of something and you subtract one of them away, you're going to have 15 of that something. And then finally, you have 14. You can view that as 14 times x to the 0, or just... | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
If you have 16 of something and you subtract one of them away, you're going to have 15 of that something. And then finally, you have 14. You can view that as 14 times x to the 0, or just 14. 14 plus 9, they're both constant terms, or they're both being multiplied by x to the 0. 14 plus 9 is 23. And we're done. Negative... | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
And the general way of writing it is y is equal to mx plus b. Where m is the slope, and here in this case m is equal to 1 3rd, so let me write that down. And b is the y intercept. So in this case b is equal to negative 2. And you know that b is the y intercept because we know that the y intercept occurs when x is equal... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
So in this case b is equal to negative 2. And you know that b is the y intercept because we know that the y intercept occurs when x is equal to 0. So if x is equal to 0 in either of these situations, this term just becomes 0 and y will be equal to b. So that's what we mean by b is the y intercept. So whenever you look ... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
So that's what we mean by b is the y intercept. So whenever you look at an equation in this form, it's actually fairly straightforward to graph this line. b is the y intercept, in this case it is negative 2. So that means that this line must intersect the y axis at y is equal to negative 2. So it's this point right her... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
So that means that this line must intersect the y axis at y is equal to negative 2. So it's this point right here. Negative 1, negative 2. This is the point 0, negative 2. If you don't believe me, there's nothing magical about this. Try evaluating or try solving for y when x is equal to 0. When x is equal to 0, this te... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
This is the point 0, negative 2. If you don't believe me, there's nothing magical about this. Try evaluating or try solving for y when x is equal to 0. When x is equal to 0, this term cancels out and you're just left with y is equal to negative 2. So that's the y intercept right there. Now, this 1 3rd tells us the slop... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
When x is equal to 0, this term cancels out and you're just left with y is equal to negative 2. So that's the y intercept right there. Now, this 1 3rd tells us the slope of the line. How much do we change in y for any change in x? So this tells us that 1 3rd, so that right there is the slope. So it tells us that 1 3rd ... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
How much do we change in y for any change in x? So this tells us that 1 3rd, so that right there is the slope. So it tells us that 1 3rd is equal to the change in y over the change in x. Or another way to think about it, if x changes by 3, then y will change by 1. So let me graph that. So we know that this point is on ... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
Or another way to think about it, if x changes by 3, then y will change by 1. So let me graph that. So we know that this point is on the graph. That's the y intercept. The slope tells us that if x changes by 3, so let me go 3 to the right, 1, 2, 3, that y will change by 1. So this must also be a point on the graph. And... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
That's the y intercept. The slope tells us that if x changes by 3, so let me go 3 to the right, 1, 2, 3, that y will change by 1. So this must also be a point on the graph. And we could keep doing that. If x changes by 3, y changes by 1. If x goes down by 3, y will go down by 1. If x goes down by 6, y will go down by 2... | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
And we could keep doing that. If x changes by 3, y changes by 1. If x goes down by 3, y will go down by 1. If x goes down by 6, y will go down by 2. It's that same ratio. So 1, 2, 3, 4, 5, 6, 1, 2. And you can see all of these points are on the line. | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
If x goes down by 6, y will go down by 2. It's that same ratio. So 1, 2, 3, 4, 5, 6, 1, 2. And you can see all of these points are on the line. And the line is the graph of this equation up here. So let me graph it. So it will look something like that. | Graph from slope-intercept equation example Algebra I Khan Academy.mp3 |
So for example, if you had the linear equation y is equal to two x plus three, that's one way to represent it, but I could represent this in an infinite number of ways. I could, let's see, I could subtract two x from both sides. I could write this as negative two x plus y is equal to three. I could manipulate it in way... | Slope-intercept form Algebra I Khan Academy.mp3 |
I could manipulate it in ways where I get it to, and I'm not gonna do it right now, but this is another way of writing that same thing. Y minus five is equal to two times x minus one. You could actually simplify this and you could get either this equation here or that equation up on top. These are all equivalent. You c... | Slope-intercept form Algebra I Khan Academy.mp3 |
These are all equivalent. You can get from one to the other with logical algebraic operations. So there's an infinite number of ways to represent a given linear equation, but what I wanna focus on in this video is this representation in particular because this one is a very useful representation of a linear equation, a... | Slope-intercept form Algebra I Khan Academy.mp3 |
And this one right over here, it's often called slope-intercept form. Slope-intercept form. And hopefully in a few minutes it will be obvious why it is called slope-intercept form. And before I explain that to you, let's just try to graph this thing. I'm gonna try to graph it. I'm just gonna plot some points here. So x... | Slope-intercept form Algebra I Khan Academy.mp3 |
And before I explain that to you, let's just try to graph this thing. I'm gonna try to graph it. I'm just gonna plot some points here. So x comma y, and I'm gonna pick some x values where it's easy to calculate the y values. So maybe the easiest is if x is equal to zero. If x is equal to zero, then two times zero is ze... | Slope-intercept form Algebra I Khan Academy.mp3 |
So x comma y, and I'm gonna pick some x values where it's easy to calculate the y values. So maybe the easiest is if x is equal to zero. If x is equal to zero, then two times zero is zero. That term goes away, and you're only left with this term right over here, y is equal to three. Y is equal to three. And so if we we... | Slope-intercept form Algebra I Khan Academy.mp3 |
That term goes away, and you're only left with this term right over here, y is equal to three. Y is equal to three. And so if we were to plot this, actually let me start plotting it. So that is my y-axis. And let me do the x-axis. So that can be my x, oh that's not as straight as I would like it. So that looks pretty g... | Slope-intercept form Algebra I Khan Academy.mp3 |
So that is my y-axis. And let me do the x-axis. So that can be my x, oh that's not as straight as I would like it. So that looks pretty good. All right, that is my x-axis. And let me mark off some hash marks here. So this is x equals one, x equals two, x equals three, this is y equals, let me do this, y equals one, y e... | Slope-intercept form Algebra I Khan Academy.mp3 |
So that looks pretty good. All right, that is my x-axis. And let me mark off some hash marks here. So this is x equals one, x equals two, x equals three, this is y equals, let me do this, y equals one, y equals two, y equals three, and obviously I can keep going, I can keep going. This would be y is equal to negative o... | Slope-intercept form Algebra I Khan Academy.mp3 |
So this is x equals one, x equals two, x equals three, this is y equals, let me do this, y equals one, y equals two, y equals three, and obviously I can keep going, I can keep going. This would be y is equal to negative one. This would be x is equal to negative one, negative two, negative three, so on and so forth. So ... | Slope-intercept form Algebra I Khan Academy.mp3 |
So this point right over here, zero comma three, this is x is zero, y is three. Well, the point that represents when x is equal to zero and y equals three, this is, we're right on the y-axis. If there of a line going through it and this line contains this point, this is going to be the y-intercept. So one way to think ... | Slope-intercept form Algebra I Khan Academy.mp3 |
So one way to think about it, the reason why this is called slope-intercept form, is it's very easy to calculate the y-intercept. The y-intercept here is going to happen when it's written in this form, it's going to happen when x is equal to zero and y is equal to three. It's going to be this point right over here. So ... | Slope-intercept form Algebra I Khan Academy.mp3 |
So it's very easy to figure out the intercept, the y-intercept from this form. Now you might be saying, oh, well it's a slope-intercept form, it must also be easy to figure out the slope from this form. And if you made that conclusion, you would be correct, and we're about to see that in a few seconds. So let's plot so... | Slope-intercept form Algebra I Khan Academy.mp3 |
So let's plot some more points here, and I'm just going to keep increasing x by one. So if you increase x by one, so we could write that our delta x, our change in x, delta Greek letter, this triangle's Greek letter delta represents change in, change in x here is one. We just increased x by one. What's going to be our ... | Slope-intercept form Algebra I Khan Academy.mp3 |
What's going to be our corresponding change in y? What's going to be our change in y? So let's see, when x is equal to one, you have two times one plus three is going to be five. So our change in y is going to be two. Let's do that again. Let's increase our x by one, change in x is equal to one. So then if we go from, ... | Slope-intercept form Algebra I Khan Academy.mp3 |
So our change in y is going to be two. Let's do that again. Let's increase our x by one, change in x is equal to one. So then if we go from, if we're going to increase by one, we're going to go from x equals one to x equals two, what's our corresponding change in y? Well, when x is equal to two, two times two is four p... | Slope-intercept form Algebra I Khan Academy.mp3 |
So then if we go from, if we're going to increase by one, we're going to go from x equals one to x equals two, what's our corresponding change in y? Well, when x is equal to two, two times two is four plus three is seven. Well, our change in y, our change in y is equal to two. We went from five, when x went from one to... | Slope-intercept form Algebra I Khan Academy.mp3 |
We went from five, when x went from one to two, y went from five to seven. So for every one that we increase x, y is increasing by two. So for this linear equation, our change in y over change in x is always going to be, our change in y is two when our change in x is one, or it's equal to two. Or we could say that our ... | Slope-intercept form Algebra I Khan Academy.mp3 |
Or we could say that our slope is equal to two. And let's just graph this to make sure that we understand this. So when x equals one, y is equal to five. And actually we're going to have to graph five up here. So when x is equal to one, y is equal to, and actually this is a little bit higher. Let me clean this up a lit... | Slope-intercept form Algebra I Khan Academy.mp3 |
And actually we're going to have to graph five up here. So when x is equal to one, y is equal to, and actually this is a little bit higher. Let me clean this up a little bit. So this one, let me erase that a little bit. Just like that. So that's y is equal to four, and this is y is equal to five. So when x is one, y is... | Slope-intercept form Algebra I Khan Academy.mp3 |
So this one, let me erase that a little bit. Just like that. So that's y is equal to four, and this is y is equal to five. So when x is one, y is equal to five. So it's that point right over there. So our line is going to look, you only need two points to define a line. Our line is going to look like, let me do this in... | Slope-intercept form Algebra I Khan Academy.mp3 |
So when x is one, y is equal to five. So it's that point right over there. So our line is going to look, you only need two points to define a line. Our line is going to look like, let me do this in this color right over here. Our line is going to look like, is going to look, is going to look something like, is going to... | Slope-intercept form Algebra I Khan Academy.mp3 |
Our line is going to look like, let me do this in this color right over here. Our line is going to look like, is going to look, is going to look something like, is going to look, let me see if I can, I didn't draw it completely at scale, but it's going to look something like this. This is the line, this is the line, y ... | Slope-intercept form Algebra I Khan Academy.mp3 |
And we already figured out that its slope is equal to two. Our change, when our change in x is one, when our change in x is one, our change in y is two. If our change in x was negative one, if our change in x was negative one, our change in y is negative two. And you could see that. If from zero we went down one, if we... | Slope-intercept form Algebra I Khan Academy.mp3 |
And you could see that. If from zero we went down one, if we went to negative one, then what's our y going to be? Two times negative one is negative two, plus three is one. So we see that the point one, or the point negative one comma one is on the line as well. So the slope here, our change in y or change in x, if we'... | Slope-intercept form Algebra I Khan Academy.mp3 |
So we see that the point one, or the point negative one comma one is on the line as well. So the slope here, our change in y or change in x, if we're going from, between any two points on this line, is always going to be two. But where did you see two in this original equation? Well, you see the two right over here. An... | Slope-intercept form Algebra I Khan Academy.mp3 |
Well, you see the two right over here. And when you write something in slope intercept form, where you explicitly solve for y, y is equal to some constant times x to the first power, plus some other constant, the second one is going to be your intercept, your y, or it's going to be a way to figure out the y intercept. ... | Slope-intercept form Algebra I Khan Academy.mp3 |
And then this two is going to represent your slope. And that makes sense, because every time you increase x by one, you're going to multiply that by two, so you're going to increase y by two. So this is just a kind of a, get your feet wet with the idea of slope intercept form, but you'll see, at least for me, this is t... | Slope-intercept form Algebra I Khan Academy.mp3 |
Because if you were given another linear equation, let's say y is equal to negative x, negative x plus two. Well, immediately you say, okay, look, my y intercept is going to be the point zero comma two, so I'm going to intersect the y axis right at that point. And then I have a slope of, the coefficient here is really ... | Slope-intercept form Algebra I Khan Academy.mp3 |
So I have a slope of negative one. So as we increase x by one, we're going to decrease y by one. Increase x by one, you're going to decrease y by one. If you increase x by two, you're going to decrease y by two. And so our line is going to look something like this. Let me see if I can draw it relatively neatly. It's go... | Slope-intercept form Algebra I Khan Academy.mp3 |
If you increase x by two, you're going to decrease y by two. And so our line is going to look something like this. Let me see if I can draw it relatively neatly. It's going to look something, it's, let me, I can do it a little bit better than that. It's because my graph paper is hand-drawn. It's not ideal. But I think ... | Slope-intercept form Algebra I Khan Academy.mp3 |
It's going to look something, it's, let me, I can do it a little bit better than that. It's because my graph paper is hand-drawn. It's not ideal. But I think you get, you get the point. It's going to look something like that. So from slope-intercept form, very easy to figure out what the y intercept is, and very easy t... | Slope-intercept form Algebra I Khan Academy.mp3 |
But I think you get, you get the point. It's going to look something like that. So from slope-intercept form, very easy to figure out what the y intercept is, and very easy to figure out the slope. The slope here, slope here is negative one. That's this negative one right over here. And the y intercept, y intercept is ... | Slope-intercept form Algebra I Khan Academy.mp3 |
So what they do over here is along the x axis, these are the inputs, and then the graph shows us what's the output. So when x is equal to seven, g of seven we see here is one. If x equals nine, g of nine here is two. If x equals six, g of six is equal to the y coordinate at this point, is equal to zero. So what is the ... | How to match function input to output given the graph (example) Algebra I Khan Academy.mp3 |
If x equals six, g of six is equal to the y coordinate at this point, is equal to zero. So what is the input value for which g of x is equal to negative two? Well, this graph right over here, this is y equals g of x. So g of x equaling negative two means y is equal to negative two. And so when does y equal negative two... | How to match function input to output given the graph (example) Algebra I Khan Academy.mp3 |
So g of x equaling negative two means y is equal to negative two. And so when does y equal negative two? Well, when does y equal negative two? It looks like that happens right at this point. And that happens when you input negative nine into g. G of negative nine is negative two. So this is going to be negative nine. A... | How to match function input to output given the graph (example) Algebra I Khan Academy.mp3 |
All right, let's work through it together. Now, when I see things in the denominator like this, my instinct is to try to not have denominators like this. And so what we could do is, to get rid of this x minus one in the denominator on the left-hand side, we can multiply both sides of the equation times x minus one. x m... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
x minus one. So we're gonna multiply both sides by x minus one. And once again, the whole point of doing that is so that we get rid of this x minus one in the denominator right over here. And then, to get rid of this x plus one in the denominator over here, we can multiply both sides of the equation times x plus one. S... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
And then, to get rid of this x plus one in the denominator over here, we can multiply both sides of the equation times x plus one. So, x plus one. Multiply both sides times x plus one. And so, what is that going to give us? Well, on the left-hand side, that is going to, x minus one divided by x minus one is just gonna ... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
And so, what is that going to give us? Well, on the left-hand side, that is going to, x minus one divided by x minus one is just gonna be one for the x's where, for the x's where that's defined, for x not being equal to one. And so, we're gonna have x plus one times negative two x plus four. So let me write that down. ... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
So let me write that down. So we have x plus, I think I'm gonna need some space, so let me make sure I don't write too big. x plus one times negative two x, negative two x plus four is going to be equal to, now, if we multiply both of these times three over x plus one, the x plus one's going to cancel with the x plus o... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
So that is going to be three x minus three, three x minus three. And then, minus, one, times both of these. So, one times x minus one, times x plus one. So minus one times x minus one times x plus one. All I did is I multiplied, took the x minus one times x plus one, multiplied it times each of these terms. When I mult... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
So minus one times x minus one times x plus one. All I did is I multiplied, took the x minus one times x plus one, multiplied it times each of these terms. When I multiplied it times this first term, the x plus one and the x plus one canceled, so I just had to multiply it three times x minus one. And then, for the seco... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
And then, for the second term, I just multiplied it times both of these. And now you might recognize this, if you have something x plus one times x minus one, that's going to be x squared minus one. So I could rewrite all of this right over here as being equal to, as being equal to x squared minus one. And once again, ... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
And once again, that's because this is the same thing as x squared minus one. And since I'm subtracting an x squared minus one, actually let me just, I don't wanna do too much on one step, so let's go to the next step. So I could multiply this out. So I could multiply x times negative two x, which would give us negativ... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
So I could multiply x times negative two x, which would give us negative two x squared, x times four, which is going to give us plus four x, and I could multiply one times negative two x, so I'm gonna subtract two x, and then one times four, which is going to be plus four, and then that is going to be equal to, that is... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
And so this is simplified to, let's see, well this is, we have a negative three and a one, so those two together are going to be equal to subtracting a two. So we can rewrite everything as, we'll do it in a neutral color now, negative two x squared plus two x plus four is equal to negative x squared plus three x, plus ... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
So let's subtract it from both sides. So we'll add x squared to both sides, add x squared, that gets rid of this white negative x squared. We subtract three x from both sides, subtract three x from both sides, add two to both sides, add two, and we will be left with, we are going to end up with, let's see, negative two... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
Two x minus three x is negative x. And then four plus two is six, is going to be equal to, well that's going to cancel with that, that, that is equal to zero. I don't like having this negative on the x squared, so let's multiply both sides times negative one. And so if I do that, so if I just take the negative of both ... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
And so if I do that, so if I just take the negative of both sides, so if I just multiply that times negative one, same thing as taking the negative of both sides, I'm going to get positive x squared plus x minus six is equal to zero. And we're making some good progress here. So we can factor this, and actually let me j... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
So if I were to factor this, what two numbers, their product is negative six, they're going to have different signs since their product is negative, and they add up to one, the coefficient on the first degree term. Well, positive three and negative two work, so I can rewrite this as x plus three times x minus two is eq... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
Yeah, three times negative two is negative six. Three x minus two x is positive x. All right, so I just factored, I just wrote this in this quadratic and factored form. And so the way that you get this equaling zero is if either one of those equals zero. x plus three equals zero, or x minus two is equal to zero. Well, ... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
And so the way that you get this equaling zero is if either one of those equals zero. x plus three equals zero, or x minus two is equal to zero. Well, this is going to happen if you subtract three from both sides, you get, that's going to happen if x is equal to negative three. Or over here, if you add two to both side... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
Or over here, if you add two to both sides, x is equal to two. So either one of these will satisfy, but we want to be careful. We want to make sure that our original equation isn't going to be undefined for either one of these. And negative three does not make either of the denominators equal to zero, so that's cool. A... | Equations with rational expressions (example 2) Mathematics III High School Math Khan Academy.mp3 |
Determine whether the points on this graph represent a function. Now, just as a refresher, a function is really just an association between members of a set that we call the domain and members of a set that we call a range. So if I take any member of the domain, let's call that x, and I give it to the function, the fun... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
So it should point to some other value. This is a function. It would not be a function if it says, well, it could point to y or it could point to z, or maybe it could point to e, or whatever else. This would not be a function because over here, so this right over here, not a function, because it's not clear if you inpu... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
This would not be a function because over here, so this right over here, not a function, because it's not clear if you input x what member of the range you're going to get. In order for it to be a function, it has to be very clear. For any input into the function, you have to be very clear that you're only going to get... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
Now, with that out of the way, let's think about this function that is defined graphically. So the ranges, or I should say the domains, the valid inputs, are the x values where this function is defined. So, for example, it tells us if x is equal to negative 1, if we assume that this over here is the x-axis and this is ... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
So one way to write that mapping is you could say x, when you input it, or let me write it this way, negative 1, if you take negative 1 and you input it into our function, I'll put a little f box right over there, you will get the number 3. This is our x and this is our y. So that seems reasonable. Negative 1, very cle... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
Negative 1, very clear that you get to 3. Let's see what happens when we go over here. If you put 2 into the function, when x is 2, y is negative 2. Once again, when x is 2, the function associates 2 for x, which is a member of the domain, it's defined for 2, it's not defined for 1. We don't know what our function is e... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
Once again, when x is 2, the function associates 2 for x, which is a member of the domain, it's defined for 2, it's not defined for 1. We don't know what our function is equal to at once, so it's not defined there. So 1 isn't part of the domain, 2 is. It tells us when x is 2, then y is going to be equal to negative 2. ... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
It tells us when x is 2, then y is going to be equal to negative 2. So it maps it or associates it with negative 2. That doesn't seem too troublesome just yet. Now let's look over here. Our function is also defined at x is equal to 3. It associates 3, our function associates or maps 3 to the value y is equal to 2. y is... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
Now let's look over here. Our function is also defined at x is equal to 3. It associates 3, our function associates or maps 3 to the value y is equal to 2. y is equal to 2. That seems pretty straightforward. Then we get to x is equal to 4, where it seems like this thing that could be a function, it is somewhat defined,... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
That seems pretty straightforward. Then we get to x is equal to 4, where it seems like this thing that could be a function, it is somewhat defined, it does try to associate 4 with things. What's interesting here is it tries to associate 4 with 2 different things. All of a sudden, in this thing that we think might have ... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
All of a sudden, in this thing that we think might have been a function, but it looks like it might not be, we don't know, do we associate 4 with 5? Do we associate it with 5? Or do we associate it with negative 1? This thing right over here is actually a relation. You can have one member of the domain being related to... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
This thing right over here is actually a relation. You can have one member of the domain being related to multiple members of the range, but if you do have that, then you're not dealing with a function. Once again, because of this, this is not a function. It's not clear that when you input 4 into it, should you output ... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
It's not clear that when you input 4 into it, should you output 5 or should you output negative 1? Sometimes there's something called the vertical line test that tells you whether something is a function. When it's graphically defined like this, you literally say, when x is 4, if I draw a vertical line, do I intersect ... | Graphical relations and functions Functions and their graphs Algebra II Khan Academy.mp3 |
The function f of x is graphed. Find f of negative 1. So this graph right over here is essentially a definition of our function. It tells us, given the allowed inputs into our function, what would the function output? So here they're saying, look, what gets output when we input x is equal to negative 1? So x equals neg... | Evaluating functions given their graph Functions and their graphs Algebra II Khan Academy.mp3 |
It tells us, given the allowed inputs into our function, what would the function output? So here they're saying, look, what gets output when we input x is equal to negative 1? So x equals negative 1 is right over here. x is equal to negative 1. And our function graph is right at 6 when f is equal to negative 1. So we c... | Evaluating functions given their graph Functions and their graphs Algebra II Khan Academy.mp3 |
It has to be perpendicular to one of the other lines. You can't be just perpendicular by yourself. And perpendicular lines, just so you have a visualization for what perpendicular lines look like. Two lines are perpendicular if they intersect at right angles. So if this is one line right there, a perpendicular line wil... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
Two lines are perpendicular if they intersect at right angles. So if this is one line right there, a perpendicular line will look like this. A perpendicular line will intersect it, but it won't just be any intersection. It will intersect at right angles. It will intersect at right angles. So these two lines are perpend... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
It will intersect at right angles. It will intersect at right angles. So these two lines are perpendicular. Now, if two lines are perpendicular, if the slope of this orange line is m, so let's say its equation is y is equal to mx plus, let's say it's b1, so it's some y-intercept, then the equation of this yellow line, ... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
Now, if two lines are perpendicular, if the slope of this orange line is m, so let's say its equation is y is equal to mx plus, let's say it's b1, so it's some y-intercept, then the equation of this yellow line, its slope is going to be the negative inverse of this guy. This guy right here is going to be y is equal to ... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
And so you could write that there. m times negative 1 over m. That's going to be, these two guys are going to cancel out, that's going to be equal to negative 1. So let's figure out the slopes of each of these lines and figure out if any of them are the negative inverse of any of the other ones. So line A, the slope is... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
So line A, the slope is pretty easy to figure out. It's already in slope-intercept form. Its slope is 3. So line A has a slope of 3. Line B, it's in standard form, not too hard to put it in slope-intercept form, so let's try to do it. So let's do line B over here. Line B, we have x plus 3y is equal to negative 21. | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
So line A has a slope of 3. Line B, it's in standard form, not too hard to put it in slope-intercept form, so let's try to do it. So let's do line B over here. Line B, we have x plus 3y is equal to negative 21. Let's subtract x from both sides so that it ends up on the right-hand side. So this, we end up with 3y is equ... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
Line B, we have x plus 3y is equal to negative 21. Let's subtract x from both sides so that it ends up on the right-hand side. So this, we end up with 3y is equal to negative x minus 21. And now let's divide both sides of this equation by 3. And we get y is equal to negative 1 third x minus 7. So this character's slope... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
And now let's divide both sides of this equation by 3. And we get y is equal to negative 1 third x minus 7. So this character's slope is negative 1 third. So here, m is equal to negative 1 third. So we already see they are the negative inverse of each other. You take the inverse of 3, it's 1 third, and then it's the ne... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
So here, m is equal to negative 1 third. So we already see they are the negative inverse of each other. You take the inverse of 3, it's 1 third, and then it's the negative of that. Or you take the inverse of negative 1 third, it's negative 3, and then this is the negative of that. So these two lines are definitely perp... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
Or you take the inverse of negative 1 third, it's negative 3, and then this is the negative of that. So these two lines are definitely perpendicular. Let's see this third line over here. So line C is 3x plus y is equal to 10. If we subtract 3x from both sides, we get y is equal to negative 3x plus 10. So our slope in t... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
So line C is 3x plus y is equal to 10. If we subtract 3x from both sides, we get y is equal to negative 3x plus 10. So our slope in this case is negative 3. So our slope here is equal to negative 3. Now this guy is the negative of that guy. This guy's slope is the negative of that, but not the negative inverse. So it's... | Perpendicular lines from equation Mathematics I High School Math Khan Academy.mp3 |
I want to make a quick clarification and then add more tools in our complex number toolkit. In the first video, I said that if I had a complex number z and it's equal to a plus bi, I used a word. And I have to be careful about that word because I use it in kind of the everyday sense. But it also has a formal reality to... | Complex conjugates Imaginary and complex numbers Precalculus Khan Academy.mp3 |
But it also has a formal reality to it. So clearly, the real part of this complex number is a. Clearly, that is the real part. And clearly, this complex number is made up of a real number plus an imaginary number. So I, just kind of talking in everyday terms, I called this the imaginary part. I called this imaginary nu... | Complex conjugates Imaginary and complex numbers Precalculus Khan Academy.mp3 |
And clearly, this complex number is made up of a real number plus an imaginary number. So I, just kind of talking in everyday terms, I called this the imaginary part. I called this imaginary number the imaginary part. But I want to just be careful there. I mean, I did make it clear that if you were to see the function,... | Complex conjugates Imaginary and complex numbers Precalculus Khan Academy.mp3 |
But I want to just be careful there. I mean, I did make it clear that if you were to see the function, the real part of z, this would spit out the a. And the function, the imaginary part of z, this would spit out. And we talked about this in the first video. It would spit out the number that's scaling the i. So it woul... | Complex conjugates Imaginary and complex numbers Precalculus Khan Academy.mp3 |
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