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Slope-intercept form Algebra I Khan Academy.mp3 | 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 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. |
Slope-intercept form Algebra I Khan Academy.mp3 | 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 this color right over here. Our line is going to look like, ... |
Slope-intercept form Algebra I Khan Academy.mp3 | 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 look, let me see if I can, I didn't draw it completely at scale, but it's ... |
Slope-intercept form Algebra I Khan Academy.mp3 | This is the line, this is the line, y is equal to two x plus three. 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 negati... |
Slope-intercept form Algebra I Khan Academy.mp3 | 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 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 poi... |
Slope-intercept form Algebra I Khan Academy.mp3 | 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're going from, between any two points on this line, is always going to be two. But where did you see two in this o... |
Slope-intercept form Algebra I Khan Academy.mp3 | But where did you see two in this original equation? 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 i... |
Slope-intercept form Algebra I Khan Academy.mp3 | The intercept itself is this point, the point at which the line intersects the y axis. 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 fee... |
Slope-intercept form Algebra I Khan Academy.mp3 | 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 the easiest form for me to think about what the graph of something looks like. Because if you were given another linear equation, let's say y is equal to negative x, negative x plus two. Well, ... |
Slope-intercept form Algebra I Khan Academy.mp3 | And then I have a slope of, the coefficient here is really just negative one. 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 lo... |
Slope-intercept form Algebra I Khan Academy.mp3 | 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 going to look something, it's, let me, I can do it a little bit better than that. It's because my g... |
Slope-intercept form Algebra I Khan Academy.mp3 | 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 you get, you get the point. It's going to look something like that. So from slope-intercept form, very easy to figu... |
Slope-intercept form Algebra I Khan Academy.mp3 | 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 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,... |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | We have a plus 5 is equal to 54. Now all this is saying is that we have some number, some variable a, and if I add 5 to it, I will get 54. And you might be able to do this in your head, but we're going to do it a little bit more systematically, because that'll be helpful for you when we do more complicated problems. So... |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | So in general, whenever you have an equation like this, we want to have the variable. We want this a all by itself on one side of the equation. We want to isolate it. It's already on the left-hand side, so let's try to get rid of everything else on the left-hand side. Well, the only other thing on the left-hand side is... |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | It's already on the left-hand side, so let's try to get rid of everything else on the left-hand side. Well, the only other thing on the left-hand side is this positive 5. Well, the best way to get rid of a plus 5 or of a positive 5 is to subtract 5. So let's subtract 5. But remember, this says a plus 5 is equal to 54. ... |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | So let's subtract 5. But remember, this says a plus 5 is equal to 54. If we want the equality to still hold, anything we do to the left-hand side of this equation, we have to do to the right side of the equation. So we also have to subtract 54 from the right. So we have a plus 5 minus 5. Well, that's just going to be a... |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | So we also have to subtract 54 from the right. So we have a plus 5 minus 5. Well, that's just going to be a plus 0, because you add 5 and you subtract 5, they cancel out. So a plus 0 is just a. And then 54 minus 5, that is 49. And we're done. We have solved for a. a is equal to 49. |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | So a plus 0 is just a. And then 54 minus 5, that is 49. And we're done. We have solved for a. a is equal to 49. And now we can check it. And we can check it by just substituting 49 back for a in our original equation. So instead of writing a plus 5 is equal to 54, let's see if 49 plus 5 is equal to 54. |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | We have solved for a. a is equal to 49. And now we can check it. And we can check it by just substituting 49 back for a in our original equation. So instead of writing a plus 5 is equal to 54, let's see if 49 plus 5 is equal to 54. So we're just substituting it back in. 49 plus 5 is equal to 54. We're trying to check t... |
Example of solving a one-step equation Linear equations Algebra I Khan Academy.mp3 | So instead of writing a plus 5 is equal to 54, let's see if 49 plus 5 is equal to 54. So we're just substituting it back in. 49 plus 5 is equal to 54. We're trying to check this. 49 plus 5 is 54. And that, indeed, is equal to 54. So it all checks out. |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | So let's do it step by step. First we're going to have this expression, negative 5 plus something. So it's going to be negative 5 plus the quantity of 4 times x. The quantity of 4 times x, well that's just going to be 4x. So it's going to be negative 5 plus 4x. So that's this expression up here. Now take the product of... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | The quantity of 4 times x, well that's just going to be 4x. So it's going to be negative 5 plus 4x. So that's this expression up here. Now take the product of negative 8. So we're going to take negative 8 and we're going to multiply the product of negative 8 and that expression. So we're going to take negative 8 and mu... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | Now take the product of negative 8. So we're going to take negative 8 and we're going to multiply the product of negative 8 and that expression. So we're going to take negative 8 and multiply it. So that expression is this thing right over here. So the product, if we say the product of negative 8 and that expression, i... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | So that expression is this thing right over here. So the product, if we say the product of negative 8 and that expression, it's going to be negative 8 times that expression. That expression is negative 5 plus 4x. So that's negative 8, that's that expression. The product of the two, so we could put a multiplication sign... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | So that's negative 8, that's that expression. The product of the two, so we could put a multiplication sign there or we could just leave that out. In implicit it would mean multiplication. Take the product of negative 8 and that expression and then add 6. So that's and then add 6. So that would be then adding 6 right o... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | Take the product of negative 8 and that expression and then add 6. So that's and then add 6. So that would be then adding 6 right over here. So we could write it as negative 8, open parentheses, negative 5 plus 4x and then add 6. Let's do one more. First consider the expression, the sum of 7 and, so that's going to be ... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | So we could write it as negative 8, open parentheses, negative 5 plus 4x and then add 6. Let's do one more. First consider the expression, the sum of 7 and, so that's going to be 7 plus something, and the product of negative 2 and x. The product of negative 2 and x is negative 2x. Negative 2x, so 7 plus negative 2x. We... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | The product of negative 2 and x is negative 2x. Negative 2x, so 7 plus negative 2x. We could write that as 7 minus 2x. So this is equal to 7 minus 2x. These are the same expression. What expression would be 4 plus, so now we're saying 4 plus, 4 plus the quantity of 2 times that expression. So it's going to be 4 plus so... |
How to write algebraic expressions with parentheses Algebra I Khan Academy.mp3 | So this is equal to 7 minus 2x. These are the same expression. What expression would be 4 plus, so now we're saying 4 plus, 4 plus the quantity of 2 times that expression. So it's going to be 4 plus some quantity, 4 plus the quantity, I'll put that in parentheses, the quantity of 2 times, do this in magenta, of 2 times... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And I encourage you to pause the video and think about it on your own. Well, there's a couple of ways to do this. One, you say, oh look, I'm multiplying two things that have the same base, so this is going to be that base, four, and then I add the exponents. Four to the negative three plus five power, which is equal to... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | Four to the negative three plus five power, which is equal to four to the second power. And that's just a straightforward exponent property, but you can also think about why does that actually make sense? Four to the negative three power, that is one over four to the third power. Or you could view that as one over four... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | Or you could view that as one over four times four times four and then four to the fifth, that's five fours being multiplied together, so it's times four times four times four times four times four. And so notice, when you multiply this out, you're gonna have five fours in the numerator and three fours in the denominat... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And so you're gonna be left with five minus three, or negative three plus five fours. So this four times four is the same thing as four squared. Now let's do one with variables. So let's say that you have a to the negative four power times a to the, let's say a squared. What is that going to be? Well, once again, you h... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | So let's say that you have a to the negative four power times a to the, let's say a squared. What is that going to be? Well, once again, you have the same base. In this case, it's a. And so, and since I'm multiplying them, you can just add the exponents. So it's gonna be a to the negative four plus two power, which is ... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | In this case, it's a. And so, and since I'm multiplying them, you can just add the exponents. So it's gonna be a to the negative four plus two power, which is equal to a to the negative two power. And once again, it should make sense. This right over here, that is one over a times a times a times a. And then this is ti... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And once again, it should make sense. This right over here, that is one over a times a times a times a. And then this is times a times a. So that cancels with that, that cancels with that, and you're still left with one over a times a, which is the same thing as a to the negative two power. Now let's do it with some qu... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | So that cancels with that, that cancels with that, and you're still left with one over a times a, which is the same thing as a to the negative two power. Now let's do it with some quotients. So what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? Well, when you're dividi... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | Well, when you're dividing, you subtract exponents if you have the same base. So this is going to be equal to 12 to the negative seven minus negative five power. You're subtracting the bottom exponent. And so this is going to be equal to 12 to the, well, subtracting a negative is the same thing as adding the positive, ... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And so this is going to be equal to 12 to the, well, subtracting a negative is the same thing as adding the positive, a 12 to the negative two power. And once again, we just have to think about why does this actually make sense? Well, you can actually rewrite this. 12 to the negative seven divided by 12 to the negative... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | 12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. If we take the reciprocal of, if we take the reciprocal of this right over here, you would make the exponent positive. And then you get exactly what we were doing in those previous... |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And so let's just do one more with variables for good measure. Let's say I have x to the negative 20th power divided by x to the fifth power. Well, once again, we have the same base and we're taking a quotient. So this is going to be x to the negative 20 minus five because we have this one right over here in the denomi... |
Linear equation word problems.mp3 | He set the temperature as high as it could go. Q represents the temperature in Quinn's home in degrees Celsius after T minutes. They say Q is equal to 15 plus 0.4 T. What was the temperature when Quinn returned from vacation? So pause this video and see if you can work this out on your own. All right, so they wanna kno... |
Linear equation word problems.mp3 | So pause this video and see if you can work this out on your own. All right, so they wanna know the temperature, and you might get a little confused, say hey, maybe T is for temperature. No, T is time in minutes. Temperature is Q. Q represents the temperature. So they really wanna know is what was Q when Quinn returned... |
Linear equation word problems.mp3 | Temperature is Q. Q represents the temperature. So they really wanna know is what was Q when Quinn returned from vacation? Well, right when Quinn returned from vacation, that is when T is equal to zero. So this is equivalent to saying what is Q, our temperature, when zero minutes have elapsed? Well, if you go back to t... |
Linear equation word problems.mp3 | So this is equivalent to saying what is Q, our temperature, when zero minutes have elapsed? Well, if you go back to this original equation, we see that Q is equal to 15 plus 0.4 times the amount of elapsed time in minutes, so that's times zero. So that's just going to be 15 degrees Celsius. If you're familiar with slop... |
Linear equation word problems.mp3 | If you're familiar with slope-intercept form, you could think of it as our temperature is equal to 0.4 times the elapsed time plus 15. So T equals zero, you're left with just this term, which in many cases we view as our y-intercept. What is going on right when we're just getting started, right when our horizontal vari... |
Linear equation word problems.mp3 | How much does the temperature increase every minute? There's a couple of ways you could think about this. One, if you recognize this as slope-intercept form, you could see that 0.4 is the slope. So that says for every one minute change in time, you're going to have an increase in temperature by 0.4 degrees Celsius. So ... |
Linear equation word problems.mp3 | So that says for every one minute change in time, you're going to have an increase in temperature by 0.4 degrees Celsius. So you could do it that way. You could try out some values. You could say, all right, let me think about what Q is going to be based on T. So time T equals zero, right when he got home, we already f... |
Linear equation word problems.mp3 | You could say, all right, let me think about what Q is going to be based on T. So time T equals zero, right when he got home, we already figured out that the temperature is 15 degrees Celsius. At T equals one, what happens? Well, it's going to be 15 plus 0.4 times one. Well, that's just going to be 15.4. Notice, when w... |
Linear equation word problems.mp3 | Well, that's just going to be 15.4. Notice, when we increased our time by one, our temperature increased by 0.4 degrees Celsius, by the slope. And it would happen again. If we increased time by another minute, if we go from one to two, we would get to 15.8. We would increase temperature by another 0.4. How much will th... |
Linear equation word problems.mp3 | If we increased time by another minute, if we go from one to two, we would get to 15.8. We would increase temperature by another 0.4. How much will the temperature increase if Quinn leaves the heat on for 20 minutes? Pause the video and see if you can answer that. All right, now we have to be careful here. They're not ... |
Linear equation word problems.mp3 | Pause the video and see if you can answer that. All right, now we have to be careful here. They're not asking us what is the temperature after 20 minutes. They're saying how much will the temperature increase if he leaves the heat on for 20 minutes. If we just want to know what is the temperature after 20 minutes, we w... |
Linear equation word problems.mp3 | They're saying how much will the temperature increase if he leaves the heat on for 20 minutes. If we just want to know what is the temperature after 20 minutes, we would just say, okay, what is Q when T is equal to 20? So it'd be 15 plus 0.4 times 20. 0.4 times 20 is eight. Eight plus 15 is 23. So it's 23 degrees Celsi... |
Linear equation word problems.mp3 | 0.4 times 20 is eight. Eight plus 15 is 23. So it's 23 degrees Celsius after 20 minutes. But that's not what they're asking us. They're asking how much will the temperature increase? Well, what did we start from? We started from 15 degrees Celsius, and now after 20 minutes, we have gone to 23 degrees Celsius so we have... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | What I want to do in this video is think about whether the product or sums of rational numbers are definitely going to be rational. So let's just first think about the product of rational numbers. So if I have one rational number, and actually let me instead of writing out the word rational, let me just represent it as... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | So I have one rational number right over there. I can represent it as a over b. And I'm going to multiply it times another rational number. And I can represent that as the ratio of two integers, m and n. And so what is this product going to be? Well, the numerator, I'm going to have am. I'm going to have a times m. And... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | And I can represent that as the ratio of two integers, m and n. And so what is this product going to be? Well, the numerator, I'm going to have am. I'm going to have a times m. And the denominator, I'm going to have b times n. b times n. Well, a is an integer, m is an integer, so you have an integer in the numerator. A... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | And b is an integer, and n is an integer, so you have an integer in the denominator. So now the product is the ratio of two integers right over here. So the product is also rational. So this thing is also rational. So if you give me the product of any two rational numbers, you're going to end up with a rational number.... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | So this thing is also rational. So if you give me the product of any two rational numbers, you're going to end up with a rational number. Let's see if the same thing is true for the sum of two rational numbers. So let's say my first rational number is a over b, and that my second can be represented as a over b. And my ... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | So let's say my first rational number is a over b, and that my second can be represented as a over b. And my second rational number can be represented as m over n. Well, how would I add these two? Well, I can find a common denominator, and the easiest one is b times n. So let me multiply this fraction, let me multiply ... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | And let me multiply this one times b in the numerator and b in the denominator. Now we've written them so they have a common denominator of bn. And so this is going to be equal to an plus bm, all of that over b times n. So b times n we've just already talked about. This is definitely going to be an integer right over h... |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | This is definitely going to be an integer right over here. And then what do we have up here? Well, we have a times n, which is an integer. b times m is another integer. The sum of two integers is going to be an integer. So you have an integer over an integer. You have the ratio of two integers. |
Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 | b times m is another integer. The sum of two integers is going to be an integer. So you have an integer over an integer. You have the ratio of two integers. So the sum of two rational numbers is going to give you another. So this one right over here was rational, and this one right over here is rational. So you take th... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | And this is a piecewise function. It's defined as essentially different lines. You see this right over here, even with all the decimals and the negative signs, this is essentially a line. It's defined by this line over this interval for x, this line over this interval of x, and this line over this interval of x. I want... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | It's defined by this line over this interval for x, this line over this interval of x, and this line over this interval of x. I want to see if we can graph it. I encourage you, especially if you have some graph paper, to see if you could graph this on your own first before I work through it. So let's think about this f... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | If when negative 10 is less than or equal to x, which is less than negative two, then our function is defined by negative 0.125x plus 4.75. So this is going to be a line, a downward-sloping line. And the easiest way I can think about graphing it is let's just plot the endpoints here and then draw the line. So when x is... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | So when x is equal to 10, so when, or sorry, when x is equal to negative 10, so we would have negative zero, actually let me write it this way. Let me do it over here where I do the, so we're gonna have negative 0.125 times negative 10 plus 4.75. That is going to be equal to, let's see, the negative times the negative ... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | That is going to be equal to six. So we're going to have the point negative 10 comma six. And that point, and it includes, so x is defined there, it's less than or equal to. And then we go all the way to negative two. So when x is equal to negative two, we have negative 0.125 times negative two plus 4.75 is equal to, s... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | And then we go all the way to negative two. So when x is equal to negative two, we have negative 0.125 times negative two plus 4.75 is equal to, see, negative times negative is positive. Two times this is going to be, is going to be positive 0.25 plus 4.75. It's going to be equal to positive five. Now, we might be temp... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | It's going to be equal to positive five. Now, we might be tempted, we might be tempted to just circle in this dot over here. But remember, this interval does not include negative two. It's up to it including, it's up to negative two, not including. So I'm gonna put a little open circle there, and then I'm gonna draw th... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | It's up to it including, it's up to negative two, not including. So I'm gonna put a little open circle there, and then I'm gonna draw the line. And then I'm gonna draw, and I'm gonna draw the line. I am going to draw my best attempt, my best attempt at the line. Now let's do the next interval. The next interval, this o... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | I am going to draw my best attempt, my best attempt at the line. Now let's do the next interval. The next interval, this one's a lot more straightforward. We start at x equals negative two. When x equals negative two, negative two plus seven is, negative two plus seven is five. So negative two, so negative two comma fi... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | We start at x equals negative two. When x equals negative two, negative two plus seven is, negative two plus seven is five. So negative two, so negative two comma five. So it actually includes that point right over there. So we're actually able to fill it in. And then when x is negative one, negative one plus seven is ... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | So it actually includes that point right over there. So we're actually able to fill it in. And then when x is negative one, negative one plus seven is going to be positive six. Positive six, but we're not including x equals negative one, up to and including. So it's going to be, it's going to be right over here. When x... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | Positive six, but we're not including x equals negative one, up to and including. So it's going to be, it's going to be right over here. When x is negative one, we are approaching, or as x approaches negative one, we're approaching negative one plus seven is six. So that's that interval right over there. And now let's ... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | So that's that interval right over there. And now let's look at this last interval. This last interval, when x is negative one, you're going to have, well this is just going to be positive 12 over 11, because we're multiplying it by negative one, plus 54 over 11, which is equal to 66 over 11, which is equal to positive... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | So we're able to fill in that right over there. And then when x is equal to 10, you have negative 120 over 11. I just multiplied this times 10. 12 times 10 is 120, and we have the negative, plus 54 over 11. So this is the same thing. This is going to be, what is this? This is negative 66 over 11. |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | 12 times 10 is 120, and we have the negative, plus 54 over 11. So this is the same thing. This is going to be, what is this? This is negative 66 over 11. Is that right? Let's see, if you, yeah, that is negative 66 over 11, which is equal to negative six. So when x is equal to 10, our function is equal to negative negat... |
Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 | This is negative 66 over 11. Is that right? Let's see, if you, yeah, that is negative 66 over 11, which is equal to negative six. So when x is equal to 10, our function is equal to negative negative six. And so this one actually doesn't have any jumps in it. It could've, but we see. So there we have it. |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | So this is x is equal to negative 5. When x is equal to negative 5, y of x is equal to 6. And when x is equal to negative 2, y of x is equal to 0. So to figure out the average rate of change of y of x with respect to x, this is going to be the change in y of x over the change of x of that interval. And the shorthand fo... |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | So to figure out the average rate of change of y of x with respect to x, this is going to be the change in y of x over the change of x of that interval. And the shorthand for change is this triangle symbol, delta. Delta y, I'll just write y, I could write delta y of x, it's delta y, change in y over our change in x. Th... |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | That's going to be our average rate of change over this interval. So how much did y change over this interval? So y went from a 6 to a 0. So let's say that we can kind of view this as our end point right over here. So this is our end, this is our start. And we could have done it the other way around, we would get a con... |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | So let's say that we can kind of view this as our end point right over here. So this is our end, this is our start. And we could have done it the other way around, we would get a consistent result. But since this is higher up on the list, let's call this the start, and the x is a lower value, we'll call that our start,... |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | But since this is higher up on the list, let's call this the start, and the x is a lower value, we'll call that our start, this is our end. So we start at 6, we end at 0. So our change in y is going to be negative 6. We went down by 6 in the y direction. It's negative 6, you could say that's 0 minus 6. And our change i... |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | We went down by 6 in the y direction. It's negative 6, you could say that's 0 minus 6. And our change in x, well we were at negative 5 and we go up to negative 2, we increased by 3. So we increased by 3. So when we increased x by 3, we decreased y of x by 6. Or if we want to simplify this right over here, negative 6 ov... |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | So we increased by 3. So when we increased x by 3, we decreased y of x by 6. Or if we want to simplify this right over here, negative 6 over 3 is the same thing as negative 2. So our average rate of change of y of x over the interval from negative 5 to negative 2 is negative 2. Every time on average x increased 1, y we... |
How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 | So our average rate of change of y of x over the interval from negative 5 to negative 2 is negative 2. Every time on average x increased 1, y went down by negative 2. |
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