Sentence
stringlengths 120
4.63k
| video_title
stringlengths 28
104
|
|---|---|
What I'd like to do in this video is a few more examples recognizing the slope and y-intercept given an equation. So let's start with something that we might already recognize. Let's say we have something in the form y is equal to five x plus three. What is the slope and the y-intercept in this example here? Well, we've already talked about that we can have something in slope-intercept form where it has the form y is equal to the slope, which people use the letter m for, the slope times x plus the y-intercept, which people use the letter b for. So if we just look at this, m is going to be the coefficient on x right over there. So m is equal to five. | Slope and y intercept from equation.mp3 |
What is the slope and the y-intercept in this example here? Well, we've already talked about that we can have something in slope-intercept form where it has the form y is equal to the slope, which people use the letter m for, the slope times x plus the y-intercept, which people use the letter b for. So if we just look at this, m is going to be the coefficient on x right over there. So m is equal to five. That is the slope. And b is just going to be this constant term, plus three. So b is equal to three. | Slope and y intercept from equation.mp3 |
So m is equal to five. That is the slope. And b is just going to be this constant term, plus three. So b is equal to three. So this is your y-intercept. So that's pretty straightforward, but let's see a few slightly more involved examples. Let's say if we had form y is equal to five plus three x. | Slope and y intercept from equation.mp3 |
So b is equal to three. So this is your y-intercept. So that's pretty straightforward, but let's see a few slightly more involved examples. Let's say if we had form y is equal to five plus three x. What is the slope and the y-intercept in this situation? Well, it might have taken you a second or two to realize how this earlier equation is different than the one I just wrote. Here, it's not five x, it's just five. | Slope and y intercept from equation.mp3 |
Let's say if we had form y is equal to five plus three x. What is the slope and the y-intercept in this situation? Well, it might have taken you a second or two to realize how this earlier equation is different than the one I just wrote. Here, it's not five x, it's just five. And this isn't three, it's three x. So if you wanna write it in the same form as we have up there, you can just swap the five and the three x, doesn't matter which one comes first, you're just adding the two. So you could rewrite it as y is equal to three x plus five. | Slope and y intercept from equation.mp3 |
Here, it's not five x, it's just five. And this isn't three, it's three x. So if you wanna write it in the same form as we have up there, you can just swap the five and the three x, doesn't matter which one comes first, you're just adding the two. So you could rewrite it as y is equal to three x plus five. And then it becomes a little bit clearer that our slope is three, the coefficient on the x term, and our y-intercept is five, y-intercept. Let's do another example. Let's say that we have the equation y is equal to 12 minus x. | Slope and y intercept from equation.mp3 |
So you could rewrite it as y is equal to three x plus five. And then it becomes a little bit clearer that our slope is three, the coefficient on the x term, and our y-intercept is five, y-intercept. Let's do another example. Let's say that we have the equation y is equal to 12 minus x. Pause this video and see if you can determine the slope and the y-intercept. All right, so something similar is going on here that we had over here. The standard form, slope-intercept form, we're used to seeing the x term before the constant term. | Slope and y intercept from equation.mp3 |
Let's say that we have the equation y is equal to 12 minus x. Pause this video and see if you can determine the slope and the y-intercept. All right, so something similar is going on here that we had over here. The standard form, slope-intercept form, we're used to seeing the x term before the constant term. So we might wanna do that over here. So we could rewrite this as y is equal to negative x plus 12, negative x plus 12. And so from this, you might immediately recognize, okay, my constant term, when it's in this form, that's my b, that is my y-intercept. | Slope and y intercept from equation.mp3 |
The standard form, slope-intercept form, we're used to seeing the x term before the constant term. So we might wanna do that over here. So we could rewrite this as y is equal to negative x plus 12, negative x plus 12. And so from this, you might immediately recognize, okay, my constant term, when it's in this form, that's my b, that is my y-intercept. So that's my y-intercept right over there. But what's my slope? Well, the slope is the coefficient on the x term. | Slope and y intercept from equation.mp3 |
And so from this, you might immediately recognize, okay, my constant term, when it's in this form, that's my b, that is my y-intercept. So that's my y-intercept right over there. But what's my slope? Well, the slope is the coefficient on the x term. But all you see is a negative here. What's the coefficient? Well, you could view negative x as the same thing as negative one x. | Slope and y intercept from equation.mp3 |
Well, the slope is the coefficient on the x term. But all you see is a negative here. What's the coefficient? Well, you could view negative x as the same thing as negative one x. So your slope here is going to be negative one. Let's do another example. Let's say that we had the equation y is equal to five x. | Slope and y intercept from equation.mp3 |
Well, you could view negative x as the same thing as negative one x. So your slope here is going to be negative one. Let's do another example. Let's say that we had the equation y is equal to five x. What's the slope and y-intercept there? At first, you might say, hey, this looks nothing like what we have up here. This is only, I only have one term on the right-hand side of the equality sign. | Slope and y intercept from equation.mp3 |
Let's say that we had the equation y is equal to five x. What's the slope and y-intercept there? At first, you might say, hey, this looks nothing like what we have up here. This is only, I only have one term on the right-hand side of the equality sign. Here, I have two. But you could just view this as five x plus zero. And then it might jump out at you that our y-intercept is zero and our slope is the coefficient on the x term. | Slope and y intercept from equation.mp3 |
This is only, I only have one term on the right-hand side of the equality sign. Here, I have two. But you could just view this as five x plus zero. And then it might jump out at you that our y-intercept is zero and our slope is the coefficient on the x term. It is equal to five. Let's do one more example. Let's say we had y is equal to negative seven. | Slope and y intercept from equation.mp3 |
And then it might jump out at you that our y-intercept is zero and our slope is the coefficient on the x term. It is equal to five. Let's do one more example. Let's say we had y is equal to negative seven. What's the slope and y-intercept there? Well, once again, you might say, hey, this doesn't look like what we had up here. How do we figure out the slope or the y-intercept? | Slope and y intercept from equation.mp3 |
Let's say we had y is equal to negative seven. What's the slope and y-intercept there? Well, once again, you might say, hey, this doesn't look like what we had up here. How do we figure out the slope or the y-intercept? Well, we could do a similar idea. We could say, hey, this is the same thing as y is equal to zero times x minus seven. And so now, it looks just like what we have over here. | Slope and y intercept from equation.mp3 |
How do we figure out the slope or the y-intercept? Well, we could do a similar idea. We could say, hey, this is the same thing as y is equal to zero times x minus seven. And so now, it looks just like what we have over here. And you might recognize that our y-intercept is negative seven y-intercept is equal to negative seven. And our slope is a coefficient on the x term. It is equal to zero. | Slope and y intercept from equation.mp3 |
So I have the function g of x is equal to 9 times 8 to the x minus 1 power. And it's defined for x being a positive, or if x is a positive integer. If x is a positive integer. So we could say the domain of this function, or all of the valid inputs here, are positive integers. So 1, 2, 3, 4, 5, on and on and on. So this is an explicitly defined function. What I now want to do is to write a recursive definition of this exact same function. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So we could say the domain of this function, or all of the valid inputs here, are positive integers. So 1, 2, 3, 4, 5, on and on and on. So this is an explicitly defined function. What I now want to do is to write a recursive definition of this exact same function. That given an x, it will give the exact same outputs. So let's first just try to understand the inputs and outputs here. So let's make a little table. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
What I now want to do is to write a recursive definition of this exact same function. That given an x, it will give the exact same outputs. So let's first just try to understand the inputs and outputs here. So let's make a little table. Let's make a table here. And let's think about what happens when we put in various x's into this function definition. So the domain is positive integers. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So let's make a little table. Let's make a table here. And let's think about what happens when we put in various x's into this function definition. So the domain is positive integers. So let's try a couple of them. 1, 2, 3, 4. And then see what the corresponding g of x is. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So the domain is positive integers. So let's try a couple of them. 1, 2, 3, 4. And then see what the corresponding g of x is. g of x. So when x is equal to 1, g of x is 9 times 8 to the 1 minus 1 power. So 9 times 8 to the 0 power, or 9 times 1. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
And then see what the corresponding g of x is. g of x. So when x is equal to 1, g of x is 9 times 8 to the 1 minus 1 power. So 9 times 8 to the 0 power, or 9 times 1. So g of x is going to be just 9. When x is 2, what's going to happen? It's going to be 9 times 8 to the 2 minus 1. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So 9 times 8 to the 0 power, or 9 times 1. So g of x is going to be just 9. When x is 2, what's going to happen? It's going to be 9 times 8 to the 2 minus 1. So that's the same thing as 9 times 8 to the 1st power. And that's just going to be 9 times 8. So that is 72. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
It's going to be 9 times 8 to the 2 minus 1. So that's the same thing as 9 times 8 to the 1st power. And that's just going to be 9 times 8. So that is 72. Actually, let me just write it that way. Let me write it as just 9 times 8. Then, when x is equal to 3, what's going on here? | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So that is 72. Actually, let me just write it that way. Let me write it as just 9 times 8. Then, when x is equal to 3, what's going on here? Well, this is going to be 3 minus 1 is 2. So it's going to be 8 squared. So it's going to be 9 times 8 squared. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
Then, when x is equal to 3, what's going on here? Well, this is going to be 3 minus 1 is 2. So it's going to be 8 squared. So it's going to be 9 times 8 squared. So we could write that as 9 times 8 times 8. I think you see a little bit of a pattern forming. When x is 4, this is going to be 8 to the 4 minus 1 power, 8 to the 3rd power. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So it's going to be 9 times 8 squared. So we could write that as 9 times 8 times 8. I think you see a little bit of a pattern forming. When x is 4, this is going to be 8 to the 4 minus 1 power, 8 to the 3rd power. So that's 9 times 8 times 8 times 8. So this gives us a good clue about how we would define this recursively. Notice, if our first term, when x equals 1, is 9, every term after that is 8 times the preceding term. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
When x is 4, this is going to be 8 to the 4 minus 1 power, 8 to the 3rd power. So that's 9 times 8 times 8 times 8. So this gives us a good clue about how we would define this recursively. Notice, if our first term, when x equals 1, is 9, every term after that is 8 times the preceding term. So let's define that as a recursive function. First, we'll define our base case. We could say g of x, and I'll do this in a new color because I'm overusing the red. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
Notice, if our first term, when x equals 1, is 9, every term after that is 8 times the preceding term. So let's define that as a recursive function. First, we'll define our base case. We could say g of x, and I'll do this in a new color because I'm overusing the red. I like the blue. g of x, well, we can define our base case. It's going to be equal to 9 if x is equal to 1. g of x equals 9 if x equals 1. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
We could say g of x, and I'll do this in a new color because I'm overusing the red. I like the blue. g of x, well, we can define our base case. It's going to be equal to 9 if x is equal to 1. g of x equals 9 if x equals 1. So that took care of that right over there. And then if it equals anything else, it equals the previous g of x. So if we're looking at, let's go all the way down to x minus 1 and then an x. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
It's going to be equal to 9 if x is equal to 1. g of x equals 9 if x equals 1. So that took care of that right over there. And then if it equals anything else, it equals the previous g of x. So if we're looking at, let's go all the way down to x minus 1 and then an x. So if this entry right over here is g of x minus 1, however many times we multiply the 8s and we have a 9 in front. So this is g of x minus 1. We know that g of x, we know that this one right over here, is going to be the previous entry, g of x minus 1, the previous entry, that's the previous entry, times 8. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So if we're looking at, let's go all the way down to x minus 1 and then an x. So if this entry right over here is g of x minus 1, however many times we multiply the 8s and we have a 9 in front. So this is g of x minus 1. We know that g of x, we know that this one right over here, is going to be the previous entry, g of x minus 1, the previous entry, that's the previous entry, times 8. So we could write that right here. So times 8. So for any other x other than 1, g of x is equal to the previous entry. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
We know that g of x, we know that this one right over here, is going to be the previous entry, g of x minus 1, the previous entry, that's the previous entry, times 8. So we could write that right here. So times 8. So for any other x other than 1, g of x is equal to the previous entry. So it's g of, I'll do that in a blue color, g of x minus 1, g of x minus 1 times 8 if x is greater than 1, or x is integer greater than 1. Now let's verify that this actually works. So let's draw another table here. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So for any other x other than 1, g of x is equal to the previous entry. So it's g of, I'll do that in a blue color, g of x minus 1, g of x minus 1 times 8 if x is greater than 1, or x is integer greater than 1. Now let's verify that this actually works. So let's draw another table here. Let's draw another table here. So once again, we're going to have x and we're going to have g of x, but this time we're going to use this recursive definition for g of x. And the reason why it's recursive is that it's referring to itself. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So let's draw another table here. Let's draw another table here. So once again, we're going to have x and we're going to have g of x, but this time we're going to use this recursive definition for g of x. And the reason why it's recursive is that it's referring to itself. In its own definition, it's saying, hey, g of x, well if x doesn't equal 1, it's going to be g of x minus 1. It's using the function itself, but we'll see that it actually does work out. So let's see, when x is equal to 1, x equals 1, so g of 1, well if x equals 1, it's equal to 9. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
And the reason why it's recursive is that it's referring to itself. In its own definition, it's saying, hey, g of x, well if x doesn't equal 1, it's going to be g of x minus 1. It's using the function itself, but we'll see that it actually does work out. So let's see, when x is equal to 1, x equals 1, so g of 1, well if x equals 1, it's equal to 9. It's equal to 9, so that was pretty straightforward. What happens when x equals 2? Well, when x equals 2, this case doesn't apply anymore. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So let's see, when x is equal to 1, x equals 1, so g of 1, well if x equals 1, it's equal to 9. It's equal to 9, so that was pretty straightforward. What happens when x equals 2? Well, when x equals 2, this case doesn't apply anymore. We go down to this case. So when x is equal to 2, it's going to be equivalent to g of 2 minus 1. Let me write this down. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
Well, when x equals 2, this case doesn't apply anymore. We go down to this case. So when x is equal to 2, it's going to be equivalent to g of 2 minus 1. Let me write this down. It's going to be equivalent to g of 2 minus 1 times 8, which is the same thing as g of 1 times 8. And what's g of 1? Well, g of 1 is right over here. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
Let me write this down. It's going to be equivalent to g of 2 minus 1 times 8, which is the same thing as g of 1 times 8. And what's g of 1? Well, g of 1 is right over here. g of 1 is 9, so this is going to be equal to 9 times 8, exactly what we got over here. And of course, this was equivalent to g of 2. So let me write this. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
Well, g of 1 is right over here. g of 1 is 9, so this is going to be equal to 9 times 8, exactly what we got over here. And of course, this was equivalent to g of 2. So let me write this. This is g of 2. I'm going to scroll over a little bit so I don't get all squenched up. So now let's go to 3. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So let me write this. This is g of 2. I'm going to scroll over a little bit so I don't get all squenched up. So now let's go to 3. Let's go to 3. And right now I'll write g of 3 first. So g of 3 is equal to, we're going to this case, it's equal to g of 3 minus 1 times 8, so that's equal to g of 2 times 8. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
So now let's go to 3. Let's go to 3. And right now I'll write g of 3 first. So g of 3 is equal to, we're going to this case, it's equal to g of 3 minus 1 times 8, so that's equal to g of 2 times 8. Well, what's g of 2? Well, g of 2, we already figured out, is 9 times 8. So it's equal to 9 times 8, that's g of 2, times 8 again. | Converting an explicit formula of a geometric sequence to a recursive formula Khan Academy.mp3 |
This is from the graph basic exponential functions on Khan Academy. And they ask us graph the following exponential function. And they give us the function, h of x is equal to 27 times 1 3rd to the x. So our initial value is 27, and 1 3rd is our common ratio. It's written in kind of standard exponential form. And they give us this little graphing tool where we can define these two points, and we can also define, we can define a horizontal asymptote to construct our function. And these three things are enough to define, to graph an exponential if we know that it is an exponential function. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
So our initial value is 27, and 1 3rd is our common ratio. It's written in kind of standard exponential form. And they give us this little graphing tool where we can define these two points, and we can also define, we can define a horizontal asymptote to construct our function. And these three things are enough to define, to graph an exponential if we know that it is an exponential function. So let's think about it a little bit. So the easiest thing that I could think of is, well, let's think about its initial value. Its initial value is going to be when x equals zero. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
And these three things are enough to define, to graph an exponential if we know that it is an exponential function. So let's think about it a little bit. So the easiest thing that I could think of is, well, let's think about its initial value. Its initial value is going to be when x equals zero. X equals zero, it's 1 3rd to the zero power, which is just one. And so you're just left with 27 times one, or just 27. That's why we call this number here when you're written it in this form. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
Its initial value is going to be when x equals zero. X equals zero, it's 1 3rd to the zero power, which is just one. And so you're just left with 27 times one, or just 27. That's why we call this number here when you're written it in this form. You call this the initial value. So when x is equal to zero, h of x is equal to 27. And we're graphing y equals h of x. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
That's why we call this number here when you're written it in this form. You call this the initial value. So when x is equal to zero, h of x is equal to 27. And we're graphing y equals h of x. So now let's graph another point. So let's think about it a little bit. When x is equal to one, when x is equal to one, what is h of x? | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
And we're graphing y equals h of x. So now let's graph another point. So let's think about it a little bit. When x is equal to one, when x is equal to one, what is h of x? Well, it's going to be 1 3rd to the first power, which is just 1 3rd. And so 1 3rd times 27 is going to be nine. So when x is one, h of one is nine. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
When x is equal to one, when x is equal to one, what is h of x? Well, it's going to be 1 3rd to the first power, which is just 1 3rd. And so 1 3rd times 27 is going to be nine. So when x is one, h of one is nine. And we can verify, well, and now let's just think about, let's think about the asymptote. So what's going to happen here when x becomes really, really, really, really, really big? Well, if I take 1 3rd to like a really large exponent, to say to the 10th power, to the 100th power, or to the 1,000th power, this thing right over here is going to start approaching zero as x becomes much, much, much, much larger. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
So when x is one, h of one is nine. And we can verify, well, and now let's just think about, let's think about the asymptote. So what's going to happen here when x becomes really, really, really, really, really big? Well, if I take 1 3rd to like a really large exponent, to say to the 10th power, to the 100th power, or to the 1,000th power, this thing right over here is going to start approaching zero as x becomes much, much, much, much larger. And so something that is approaching zero times 27, well, that's going to approach zero as well. So we're going to have a horizontal asymptote at zero. And you can verify that this works for more than just the two points we thought about. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
Well, if I take 1 3rd to like a really large exponent, to say to the 10th power, to the 100th power, or to the 1,000th power, this thing right over here is going to start approaching zero as x becomes much, much, much, much larger. And so something that is approaching zero times 27, well, that's going to approach zero as well. So we're going to have a horizontal asymptote at zero. And you can verify that this works for more than just the two points we thought about. When x is equal to two, this is telling us that the graph y equals h of x goes through the point two comma three. So h of two should be equal to three. And you can verify that that is indeed the case. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
And you can verify that this works for more than just the two points we thought about. When x is equal to two, this is telling us that the graph y equals h of x goes through the point two comma three. So h of two should be equal to three. And you can verify that that is indeed the case. If x is two, 1 3rd squared is nine, oh, sorry, 1 3rd squared is 1 9th, times 27 is three. And we see that right over here. When x is two, h of two is three. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
And you can verify that that is indeed the case. If x is two, 1 3rd squared is nine, oh, sorry, 1 3rd squared is 1 9th, times 27 is three. And we see that right over here. When x is two, h of two is three. So I feel pretty good about that. Let's do another one of these. So graph the following exponential function. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
When x is two, h of two is three. So I feel pretty good about that. Let's do another one of these. So graph the following exponential function. So same logic. When x is zero, the g of zero is just going to boil down to that initial value. And so let me scroll down. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
So graph the following exponential function. So same logic. When x is zero, the g of zero is just going to boil down to that initial value. And so let me scroll down. The initial value is negative 30. And so let's think about when x is equal to one. When x is equal to one, two to the 1st power is just two. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
And so let me scroll down. The initial value is negative 30. And so let's think about when x is equal to one. When x is equal to one, two to the 1st power is just two. And so two times negative 30 is negative 60. So when x is equal to one, the value of the graph is negative 60. Now let's think about this asymptote where that should sit. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
When x is equal to one, two to the 1st power is just two. And so two times negative 30 is negative 60. So when x is equal to one, the value of the graph is negative 60. Now let's think about this asymptote where that should sit. So let's think about what happens when x becomes really, really, really, really, really, really negative. So when x is really negative, so two to the negative one power is 1 1st, two to the negative two is 1 4th, two to the negative three is 1 8th. As you get larger and larger negative, or higher magnitude negative values, or in other words, as x becomes more and more and more negative, two to that power is going to approach zero. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
Now let's think about this asymptote where that should sit. So let's think about what happens when x becomes really, really, really, really, really, really negative. So when x is really negative, so two to the negative one power is 1 1st, two to the negative two is 1 4th, two to the negative three is 1 8th. As you get larger and larger negative, or higher magnitude negative values, or in other words, as x becomes more and more and more negative, two to that power is going to approach zero. And so negative 30 times something approaching zero is going to approach zero. So this asymptote's in the right place. Our horizontal asymptote, as x approaches negative infinity, as we move further and further to the left, the value of the function is going to approach zero. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
As you get larger and larger negative, or higher magnitude negative values, or in other words, as x becomes more and more and more negative, two to that power is going to approach zero. And so negative 30 times something approaching zero is going to approach zero. So this asymptote's in the right place. Our horizontal asymptote, as x approaches negative infinity, as we move further and further to the left, the value of the function is going to approach zero. And we can see it kind of approaches zero from below. We can see that it approaches below because we already looked at the initial value and we used that common ratio to find one other point. Hopefully you found that interesting. | Graphing exponential growth & decay Mathematics I High School Math Khan Academy.mp3 |
And our goal in this video is to figure out at what x value, so when, when does f of x, so at what x value is f of x going to be equal to negative 1 25th? And you might be tempted to just eyeball it over here, but when f of x is negative 1 25th, that's like right below the x axis. So if I tried to, if I tried to eyeball it, it would be very difficult. It's very difficult to tell what value that is. It might be at three, it might be at four. I am not sure. So instead of, actually it looks like, no, maybe, well, I don't wanna just eyeball it, just guess it. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
It's very difficult to tell what value that is. It might be at three, it might be at four. I am not sure. So instead of, actually it looks like, no, maybe, well, I don't wanna just eyeball it, just guess it. Instead, I'm gonna actually find an expression that defines f of x, because they've given us some information here, and then I can just solve for x. So let's do that. Well, since we know that f of x is an exponential function, we know it's gonna take the form f of x is equal to our initial value, a, times our common ratio, r, to the xth power. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
So instead of, actually it looks like, no, maybe, well, I don't wanna just eyeball it, just guess it. Instead, I'm gonna actually find an expression that defines f of x, because they've given us some information here, and then I can just solve for x. So let's do that. Well, since we know that f of x is an exponential function, we know it's gonna take the form f of x is equal to our initial value, a, times our common ratio, r, to the xth power. Well, the initial value is straightforward enough. That's going to be the value that the function takes on when x is equal to zero. And you can even see it here. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Well, since we know that f of x is an exponential function, we know it's gonna take the form f of x is equal to our initial value, a, times our common ratio, r, to the xth power. Well, the initial value is straightforward enough. That's going to be the value that the function takes on when x is equal to zero. And you can even see it here. If x is equal to zero, the r to the x would just be one, and so f of zero will just be equal to a. And so what is f of zero? Well, when x is equal to zero, this is essentially we're saying where does it intersect, where does it intersect the y-axis? | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
And you can even see it here. If x is equal to zero, the r to the x would just be one, and so f of zero will just be equal to a. And so what is f of zero? Well, when x is equal to zero, this is essentially we're saying where does it intersect, where does it intersect the y-axis? We see f of zero is negative 25. So a is going to be negative 25. Negative 25. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Well, when x is equal to zero, this is essentially we're saying where does it intersect, where does it intersect the y-axis? We see f of zero is negative 25. So a is going to be negative 25. Negative 25. When x is zero, the r to the x is just one, so f of zero is going to be negative 25. We see that right over there. Now, to figure out the common ratio, there's a couple of ways you could think about it. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Negative 25. When x is zero, the r to the x is just one, so f of zero is going to be negative 25. We see that right over there. Now, to figure out the common ratio, there's a couple of ways you could think about it. The common ratio is the ratio between two successive values that are separated by one. What do I mean by that? Well, you could view it as the ratio between f of one and f of zero. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Now, to figure out the common ratio, there's a couple of ways you could think about it. The common ratio is the ratio between two successive values that are separated by one. What do I mean by that? Well, you could view it as the ratio between f of one and f of zero. That's going to be the common ratio, or the ratio between f of two and f of one. That is going to be the common ratio. Well, lucky for us, we know f of zero is negative 25, and we know that f of one, f of one, x is equal to one, y, or f of x, or f of one is equal to negative five. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Well, you could view it as the ratio between f of one and f of zero. That's going to be the common ratio, or the ratio between f of two and f of one. That is going to be the common ratio. Well, lucky for us, we know f of zero is negative 25, and we know that f of one, f of one, x is equal to one, y, or f of x, or f of one is equal to negative five. Negative five. And so just like that, we're able to figure out that our common ratio r is negative five over negative 25, which is the same thing as 1 5th. Divide a negative by a negative, you get a positive. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Well, lucky for us, we know f of zero is negative 25, and we know that f of one, f of one, x is equal to one, y, or f of x, or f of one is equal to negative five. Negative five. And so just like that, we're able to figure out that our common ratio r is negative five over negative 25, which is the same thing as 1 5th. Divide a negative by a negative, you get a positive. So you're gonna have five over 25, which is 1 5th. Which is 1 5th. So now we can write, we can write an expression that defines f of x. f of x is going to be equal to negative 25 times, times 1 5th to the x power. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Divide a negative by a negative, you get a positive. So you're gonna have five over 25, which is 1 5th. Which is 1 5th. So now we can write, we can write an expression that defines f of x. f of x is going to be equal to negative 25 times, times 1 5th to the x power. And so let's go back to our question. When is this going to be equal to negative 1 25th? So when does this equal to negative 1 25th? | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
So now we can write, we can write an expression that defines f of x. f of x is going to be equal to negative 25 times, times 1 5th to the x power. And so let's go back to our question. When is this going to be equal to negative 1 25th? So when does this equal to negative 1 25th? Well, let's just set them equal to each other. So let, there's a siren outside, I don't know if you hear it. So negative, I'll power through. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
So when does this equal to negative 1 25th? Well, let's just set them equal to each other. So let, there's a siren outside, I don't know if you hear it. So negative, I'll power through. All right, negative, so let's see, at what x value does this expression equal negative 1 25th? Let's see, we can multiply, well actually, we wanna solve for x, so let's see, let's divide both sides by negative 25. And so we are going to get 1 5th to the x power is equal to, if we divide both sides by negative 25, this negative 25 is gonna go away, and on the right hand side, we're going to have, divided negative by negative is gonna be positive, it's gonna be 1 over 625. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
So negative, I'll power through. All right, negative, so let's see, at what x value does this expression equal negative 1 25th? Let's see, we can multiply, well actually, we wanna solve for x, so let's see, let's divide both sides by negative 25. And so we are going to get 1 5th to the x power is equal to, if we divide both sides by negative 25, this negative 25 is gonna go away, and on the right hand side, we're going to have, divided negative by negative is gonna be positive, it's gonna be 1 over 625. It's gonna be 1 over 625. And 1 5th to the x power, this is the same thing as 1 to the x power over 5 to the x power is equal to, is equal to 1 over 625. 1 over 625. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
And so we are going to get 1 5th to the x power is equal to, if we divide both sides by negative 25, this negative 25 is gonna go away, and on the right hand side, we're going to have, divided negative by negative is gonna be positive, it's gonna be 1 over 625. It's gonna be 1 over 625. And 1 5th to the x power, this is the same thing as 1 to the x power over 5 to the x power is equal to, is equal to 1 over 625. 1 over 625. Well, 1 to the x power is just going to be equal to, is just going to be equal to 1. So we could really, it doesn't matter that we have this to the x power over here. And so we can see, I thought I was erasing that with a black color, there you go. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
1 over 625. Well, 1 to the x power is just going to be equal to, is just going to be equal to 1. So we could really, it doesn't matter that we have this to the x power over here. And so we can see, I thought I was erasing that with a black color, there you go. That's a black color right over there. So we can see that 5 to the x power needs to be equal to 625. So let me write that over here. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
And so we can see, I thought I was erasing that with a black color, there you go. That's a black color right over there. So we can see that 5 to the x power needs to be equal to 625. So let me write that over here. 5, whoops, didn't change my color. 5 to the x power needs to be equal to 625. Now, the best way I could think of doing this is let's just think about our powers of 5. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
So let me write that over here. 5, whoops, didn't change my color. 5 to the x power needs to be equal to 625. Now, the best way I could think of doing this is let's just think about our powers of 5. So 5 to the 1st power is 5. 5 squared is 25. 5 to the 3rd is 125. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Now, the best way I could think of doing this is let's just think about our powers of 5. So 5 to the 1st power is 5. 5 squared is 25. 5 to the 3rd is 125. 5 to the 4th, we'll multiply that by 5, you're gonna get 625. So x is going to be 4. Is going to be 4, because 5 to the 4th power is 625. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
5 to the 3rd is 125. 5 to the 4th, we'll multiply that by 5, you're gonna get 625. So x is going to be 4. Is going to be 4, because 5 to the 4th power is 625. So we can now say that f of 4, f of 4 is equal to, is equal to negative 1 25th, is equal to negative 1 25th. And once again, you can verify that. You can verify that right over here. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
Is going to be 4, because 5 to the 4th power is 625. So we can now say that f of 4, f of 4 is equal to, is equal to negative 1 25th, is equal to negative 1 25th. And once again, you can verify that. You can verify that right over here. 1 5th to, 1 5th to the 4th power is gonna be 1 over 625. 25, negative 25 over positive 625 is gonna be negative 1 25th. So hopefully that clears things up a little bit. | Analyzing graphs of exponential functions negative initial value High School Math Khan Academy.mp3 |
And let's say that Diya, Diya today, is two, two years old. And what I am curious about in this video is how many years will it take? And let me write this down. How many years will it take? Will it take, will it take for Arman, for Arman to be three times, times as old as Diya? As Diya. So that's the question right there and I encourage you to try to take a shot at this yourself. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
How many years will it take? Will it take, will it take for Arman, for Arman to be three times, times as old as Diya? As Diya. So that's the question right there and I encourage you to try to take a shot at this yourself. So let's think about this a little bit. We're asking how many years will it take? That's what we don't know. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
So that's the question right there and I encourage you to try to take a shot at this yourself. So let's think about this a little bit. We're asking how many years will it take? That's what we don't know. That's what we're curious about. How many years will it take for Arman to be three times as old as Diya? So let's set, let's set some variable. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
That's what we don't know. That's what we're curious about. How many years will it take for Arman to be three times as old as Diya? So let's set, let's set some variable. Let's say y for years. Let's say y is equal to years, years it will take. So given that, can we now set up an equation, given this information, to figure out how many years it will take for Arman to be three times as old as Diya? | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
So let's set, let's set some variable. Let's say y for years. Let's say y is equal to years, years it will take. So given that, can we now set up an equation, given this information, to figure out how many years it will take for Arman to be three times as old as Diya? Well, let's think about how, how old Arman will be in y years. How old will he be? Well, in y years, let me write it here. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
So given that, can we now set up an equation, given this information, to figure out how many years it will take for Arman to be three times as old as Diya? Well, let's think about how, how old Arman will be in y years. How old will he be? Well, in y years, let me write it here. In, in y years, Arman, Arman, Arman is going to be how old? Arman is going to be, well he's 18 right now, and in y years, he's going to be y years older. So in y years, Arman is going to be 18 plus y. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
Well, in y years, let me write it here. In, in y years, Arman, Arman, Arman is going to be how old? Arman is going to be, well he's 18 right now, and in y years, he's going to be y years older. So in y years, Arman is going to be 18 plus y. And what about Diya? How old will she be? How old will she be in y years? | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
So in y years, Arman is going to be 18 plus y. And what about Diya? How old will she be? How old will she be in y years? Well, she's two now, and in y years, she'll just be two plus y. So what we're curious about, now that we know this, is how many years will it take for this quantity, for this expression, to be three times this quantity? So we're really curious. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
How old will she be in y years? Well, she's two now, and in y years, she'll just be two plus y. So what we're curious about, now that we know this, is how many years will it take for this quantity, for this expression, to be three times this quantity? So we're really curious. We want to solve for y, such that 18 plus y is going to be equal to three times, is going to be equal to three times two plus y. Three times two plus y. Notice, this is Arman in y years, this is Diya in y years, and we're saying that what Arman's going to be in y years is three times what Diya is going to be in y years. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
So we're really curious. We want to solve for y, such that 18 plus y is going to be equal to three times, is going to be equal to three times two plus y. Three times two plus y. Notice, this is Arman in y years, this is Diya in y years, and we're saying that what Arman's going to be in y years is three times what Diya is going to be in y years. So we've set up our equation, now we can just solve it. So let's take this step by step. So on the left-hand side, and maybe I'll do this in a new color, just so I don't have to keep switching. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
Notice, this is Arman in y years, this is Diya in y years, and we're saying that what Arman's going to be in y years is three times what Diya is going to be in y years. So we've set up our equation, now we can just solve it. So let's take this step by step. So on the left-hand side, and maybe I'll do this in a new color, just so I don't have to keep switching. So on the left-hand side, I still have 18 plus y, and on the right-hand side, I can distribute this three. So three times two is six, three times y is three y. Six plus three y. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
So on the left-hand side, and maybe I'll do this in a new color, just so I don't have to keep switching. So on the left-hand side, I still have 18 plus y, and on the right-hand side, I can distribute this three. So three times two is six, three times y is three y. Six plus three y. And then it's always nice to get all of our constants on one side of the equation, and all of our variables on the other side of the equation. So we have a three y over here. We have more y's on the right-hand side than the left-hand side. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
Six plus three y. And then it's always nice to get all of our constants on one side of the equation, and all of our variables on the other side of the equation. So we have a three y over here. We have more y's on the right-hand side than the left-hand side. So let's get rid of the y's on the left-hand side. You could do it either way, but you'd end up with negative numbers. So let's subtract a y from each side. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
We have more y's on the right-hand side than the left-hand side. So let's get rid of the y's on the left-hand side. You could do it either way, but you'd end up with negative numbers. So let's subtract a y from each side. And we are left with, on the left-hand side, 18. And on the right-hand side, you have six plus three y's. Take away one of those y's, you're going to be left with two y's. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
So let's subtract a y from each side. And we are left with, on the left-hand side, 18. And on the right-hand side, you have six plus three y's. Take away one of those y's, you're going to be left with two y's. Now we can get rid of the constant term here. So we will subtract six from both sides. Subtract six from both sides. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
Take away one of those y's, you're going to be left with two y's. Now we can get rid of the constant term here. So we will subtract six from both sides. Subtract six from both sides. 18 minus six is 12. The whole reason why we subtracted six from the right was to get rid of this. Six minus six is zero, so you have 12 is equal to two y. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
Subtract six from both sides. 18 minus six is 12. The whole reason why we subtracted six from the right was to get rid of this. Six minus six is zero, so you have 12 is equal to two y. Two times the number of years it will take is 12. And you could probably solve this in your head, but if we just want a one coefficient here, we would divide by two on the right. Whatever we do to one side of the equation, we have to do it on the other side, or the equation will not still be an equation. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
Six minus six is zero, so you have 12 is equal to two y. Two times the number of years it will take is 12. And you could probably solve this in your head, but if we just want a one coefficient here, we would divide by two on the right. Whatever we do to one side of the equation, we have to do it on the other side, or the equation will not still be an equation. So we are left with y is equal to six. Or y is equal to six. So, going back to the question, how many years will it take for Armand to be three times as old as Dia? | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
Whatever we do to one side of the equation, we have to do it on the other side, or the equation will not still be an equation. So we are left with y is equal to six. Or y is equal to six. So, going back to the question, how many years will it take for Armand to be three times as old as Dia? Well, it's going to take six years. Now, I want you to verify this. Think about it. | Ex 1 age word problem Linear equations Algebra I Khan Academy.mp3 |
End of preview. Expand
in Dataset Viewer.
No dataset card yet
New: Create and edit this dataset card directly on the website!
Contribute a Dataset Card- Downloads last month
- 0