Slope-intercept equation from graph (video) | Khan Academy

Video transcript

So you may or may not already know that any linear equation can be written in the form yttrium is equal to mx plus b. Where thousand is the slope of the credit line. The like slope that we’ve been dealing with the final few videos. The wax over run of the credit line. Or the dip of the wrinkle. And barn is the y-intercept. I think it ‘s pretty easy to verify that b is a y-intercept. The way you verify that is you substitute x is equal to 0. If you get adam is equal to 0 — remember ten is equal to 0, that means that ‘s where we ‘re going to intercept at the y-axis. If ten is equal to 0, this equation becomes y is peer to m times 0 plus b. thousand times 0 is barely going to be 0. I do n’t care what m is. so then y is going to be equal to b. So the sharpen 0, b-complex vitamin is going to be on that line. The line will intercept the y-axis at the point y is peer to b. We ‘ll see that with actual numbers in the following few videos. Just to verify for you that m is actually the gradient, let ‘s equitable try some numbers out. We know the target 0, bacillus is on the line. What happens when ten is adequate to 1 ? You get y is equal to m times 1. Or it ‘s equal to m summation barn. So we besides know that the point 1, thousand plus bacillus is besides on the tune. right ? This is barely the y value. So what ‘s the slope between that point and that point ? Let ‘s take this as the end point, so you have m plus b-complex vitamin, our change in y, megabyte plus bacillus subtraction bel over our change in ten, over 1 minus 0. This is our change in yttrium over change in adam. We ‘re using two points. That ‘s our end point. That ‘s our start point. So if you simplify this, bacillus minus b is 0. 1 minus 0 is 1. so you get m/1, or you get it ‘s peer to m. So hopefully you ‘re satisfied and hopefully I did n’t confuse you by stating it in the abstract with all of these variables hera. But this is decidedly going to be the gradient and this is decidedly going to be the y-intercept. now given that, what I want to do in this exercise is look at these graphs and then use the already drawn graph to figure out the equation. So we ‘re going to look at these, figure out the slopes, figure out the y-intercepts and then know the equality. So let ‘s do this line A foremost. sol what is A ‘s slope ? Let ‘s start at some arbitrary point. Let ‘s start right over there. We want to get evening numbers. If we run one, two, three. so if delta ten is equal to 3. veracious ? One, two, three. Our delta yttrium — and I ‘m just doing it because I want to hit an even number here — our delta yttrium is equal to — we go down by 2 — it ‘s equal to negative 2. thus for A, change in y for deepen in ten. When our change in adam is 3, our change in yttrium is negative 2. sol our slope is negative 2/3. When we go over by 3, we’re going to go down by 2. Or if we go over by 1, we’re going to go down by 2/3. You ca n’t precisely see it there, but you decidedly see it when you go over by 3. So that ‘s our gradient. We ‘ve basically done one-half of that problem. nowadays we have to figure out the y-intercept. So that right there is our m. immediately what is our barn ? Our y-intercept. Well where does this intersect the y-axis ? Well we already said the gradient is 2/3. so this is the period y is peer to 2. When we go over by 1 to the right, we would have gone gloomy by 2/3. so this right here must be the point 1 1/3. Or another means to say it, we could say it ‘s 4/3. That ‘s the compass point y is peer to 4/3. correct there. A little bit more than 1. About 1 1/3. So we could say b is peer to 4/3. So we ‘ll know that the equation is yttrium is equal to m, damaging 2/3, x plus b-complex vitamin, plus 4/3. That ‘s equation A. Let ‘s do equation B. Hopefully we wo n’t have to deal with equally many fractions here. equality B. Let ‘s figure out its slope first gear. Let ‘s start at some reasonable point. We could start at that point. Let me do it right here. B. equality B. When our delta ten is equal to — let me write it this manner, delta x. indeed our delta ten could be 1. When we move over 1 to the right, what happens to our delta yttrium ? We go up by 3. delta x. delta yttrium. Our change in yttrium is 3. so delta y over delta x, When we go to the right, our change in ten is 1. Our change in y is positive 3. thus our slope is equal to 3. What is our y-intercept ? Well, when adam is peer to 0, y is equal to 1. thus b is equal to 1. so this was a batch easier. hera the equation is yttrium is peer to 3x plus 1. Let ‘s do that stopping point line there. Line C Let ‘s do the y-intercept first. You see immediately the y-intercept — when adam is adequate to 0, yttrium is negative 2. so bacillus is peer to negative 2. And then what is the slope ? m is peer to change in y over variety in x. Let ‘s start at that y-intercept. If we go over to the correctly by one, two, three, four. so our change in ten is adequate to 4. What is our change in y ? Our change in yttrium is positive 2. therefore change in y is 2 when switch in ten is 4. So the gradient is adequate to 1/2, 2/4. So the equation here is y is adequate to 1/2 x, that ‘s our slope, minus 2. And we ‘re done. now let ‘s go the other way. Let ‘s spirit at some equations of lines knowing that this is the slope and this is the y-intercept — that ‘s the molarity, that ‘s the barn — and actually graph them. Let ‘s do this first production line. I already started circling it in orange. The y-intercept is 5. When ten is equal to 0, y is adequate to 5. You can verify that on the equality. therefore when ten is equal to 0, y is peer to one, two, three, four, five. That ‘s the y-intercept and the slope is 2. That means when I move 1 in the x-direction, I move up 2 in the y-direction. If I move 1 in the x-direction, I move up 2 in the y-direction. If I move 1 in the x-direction, I move up 2 in the y-direction. If I move rear 1 in the x-direction, I move down 2 in the y-direction. If I move back 1 in the x-direction, I move down 2 in the y-direction. I keep doing that. so this line is going to look — I ca n’t draw lines excessively neatly, but this is going to be my best guess. It ‘s going to look something like that. It ‘ll just keep going on, on and on and on. So that ‘s our first production line. I can just keep going down like that. Let ‘s do this moment line. y is equal to negative 0.2x plus 7. Let me write that. yttrium is equal to negative 0.2x plus 7. It ‘s always easier to think in fractions. therefore 0.2 is the lapp thing as 1/5. We could write y is equal to damaging 1/5 ten plus 7. We know it ‘s y-intercept at 7. So it ‘s one, two, three, four, five, six. That ‘s our y-intercept when ten is equal to 0. This tells us that for every 5 we move to the right, we move down 1. We can view this as negative 1/5. The delta y over delta ten is equal to negative 1/5. For every 5 we move to the right, we move down 1. therefore every 5. One, two, three, four, five. We moved 5 to the right. That means we must move depressed 1. We move 5 to the right. One, two, three, four, five. We must move polish 1. If you go backwards, if you move 5 backwards — rather of this, if you view this as 1 over negative 5. These are obviously equivalent numbers. If you go bet on 5 — that’s negative 5. One, two, three, four, five. then you move up 1. If you go rear 5 — one, two, three, four, five — you move up 1. So the agate line is going to look like this. I have to just connect the dots. I think you get the mind. I just have to connect those dots. I could ‘ve drawn it a little bit straighter. now let ‘s do this one, y is equal to negative adam. Where ‘s the b terminus ? I do n’t see any barn term. You remember we ‘re saying yttrium is adequate to mx plus b. Where is the b-complex vitamin ? Well, the b is 0. You could view this as plus 0. hera is b-complex vitamin is 0. When ten is 0, yttrium is 0. That ‘s our y-intercept, right there at the origin. And then the slope — once again you see a negative signal. You could view that a negative 1x plus 0. So slope is minus 1. When you move to the justly by 1, when change in x is 1, change in yttrium is negative 1. When you move up by 1 in ten, you go polish by 1 in y. Or if you go down by 1 in ten, you ‘re going to go up by 1 in y. x and y are going to have opposite signs. They go in reverse directions. So the line is going to look like that. You could about imagine it’s splitting the second and fourthly quadrants. now I ‘ll do one more. Let ‘s do this last one proper here. y is equal to 3.75. now you ‘re saying, g, we ‘re looking for y is equal to mx plus b. Where is this ten term ? It ‘s completely gone. well the reality hera is, this could be rewritten as y is peer to 0x plus 3.75. now it makes sense. The gradient is 0. No matter how much we change our ten, y does not change. Delta yttrium over delta x is equal to 0. I do n’t care how much you change your x. Our y-intercept is 3.75. so 1, 2, 3.75 is good around there. You want to get close. 3 3/4. As I change ten, y will not change. y is constantly going to be 3.75. It ‘s just going to be a horizontal cable at y is equal to 3.75. anyhow, hopefully you found this utilitarian.

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