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CCSS.Math:

in this video I'm going to do a bunch of example slope slope problems and just as a bit of review slope is just a way of measuring the inclination of a line and the definition we're going to hopefully get a good working knowledge of it in this video the definition of it is change in Y divided by change in X and this may or may not make some sense to you right now but as we do more and more examples I think it'll make a good amount of sense let's do this first line right here line a let's figure out its slope and they've actually drawn two points here that we can use as the reference points so first of all let's look at the coordinates of those points so you have this point right here what's its coordinates its x-coordinate is 3 and its y-coordinate is 6 and then down here this point x-coordinate is negative 1 and it's y-coordinate is negative 6 so there's a couple of ways we can think about slope one is we could look at it straight up using the formula we could say change in Y so slope is change in Y over change in X and we can figure it out numerically I know in a second draw it graphically so what's our change in Y our change in Y is literally how much did our y-values change going from this point to that point so how much did our y-values change our Y we went from here why is it negative 6 and it went all the way up to all the way up to positive 6 so what's this distance right here well it's going to be your end point Y value it's going to be 6 minus your starting point Y value minus negative 6 or 6 plus 6 which is equal to 12 and you could just count this you say 1 2 3 4 5 6 7 8 9 10 11 12 so when we changed our Y value by 12 we had to change our x value by what was our change in x over the same change in Y well we went from X is equal to negative 1 to X is equal to three right X went from negative one to three so we do the endpoint which is three minus the starting point which is negative one which is equal to four so our change in Y over change in X is equal to 12 over 4 or if we want to write this in simplest form this is the same thing as three now the interpretation of this means that for every one we move over we could view this let me write it this way change in Y over change in X is equal to we could say it's three or we could say it's three over one which tells us that for every one we move in the positive x direction we're going to move up three because this is a positive 3 in the Y direction you can see that when we moved one in the X we moved up three and the y when we moved one in the X we moved up three and the y if you move two in the x direction you're going to move six in the y 6 over 2 is the same thing as three so this 3 tells us how quickly do we go up as we increase X let's do the same thing for the second line on this graph graph B same idea and I'm going to use the points that they gave us but you could use any points on that line so let's see we have one point here which is the point 0 comma 1 you have 0 comma 1 and then the starting point we could call this the finish point the starting point right here we could view it as let's see X is negative 6 negative 6 and Y is negative 2 so same idea what is the change in Y given some change in X and so let's let's do the change in X first so what is our change in X so in this situation what is our change in X Delta X we could even counted it's 1 2 3 4 5 6 it's going to be 6 but if you wouldn't have a graph to count from you could literally take your you could take your finishing X position so it's 0 and subtract from that you're starting x position 0 minus negative 6 so when you're changing is equal to so this will be six what is our change in Y and remember we're taking this as our finishing position that's our finishing position this is our starting position so we took zero minus negative six so then on the Y we have to do 1 minus negative 2 1 minus negative 2 so what's 1 minus negative 2 that's the same thing as 1 plus 2 that is equal to 3 so it is 3/6 or 1/2 so notice when our when removed in the x-direction by 6 we moved in the y-direction by positive 3 so our change in Y was 3 when our change in X was 6 now one of the things that confuses a lot of people is how do I know what order to to you know how did I know to do the 0 first and the negative 6 second and then the 1 first and then the negative 2 second and the answer is you could have done it in either order as long as you keep them straight so you could have also have done change in Y over change in X we could have said it's equal to negative 2 minus 1 so negative 2 minus 1 so we're using this coordinate first negative 2 minus 1 for the Y over negative 6 negative 6 minus 0 notice this this is the negative of that that is the negative of that but since we have a negative over negative they're going to cancel out so this is going to be equal to negative 3 over negative 6 the negatives cancel out this is also equal to 1/2 so the important thing is is if you use this if you use this x coordinate sorry if you use this y coordinate first then you so we use this y coordinate first then you have to use this x coordinate first as well if you use this y coordinate first as we did here then you have to use this x coordinate first as you did there you just have to make sure that your change in X and change in Y are you're using the same final and starting points and just to interpret this what's this this is saying this is saying that for every minus 6 we go and X so if we go down if we go minus 6 and X so that's going backwards we're going to go - 3 + y -3 and Y but they're essentially same saying the same thing the slope of this line is 1/2 which tells us for every 2 we travel in X we go up 1 and Y or if we go back to an X we go down 1 and Y that's what 1/2 slope tells us and notice the line with a 1/2 slope it is less steep than the line with a slope of 3 let's do a couple of more of these so let's do line let's do this line see right here I'll do it in pink and let's say that the starting point I'm just picking this arbitrarily the well I'm using these points that they've drawn here the starting point is that the coordinate negative 1 6 and that my finishing point is at the point what is this 5 negative 6 5 negative 6 so we could do our slope is going to be let me write this slope is going to be equal to change in X so I change in Y I'll never forget that change in Y over change in X sometimes it said rise over run run is how much you're moving in the horizontal direction rise is how much you're moving in the vertical direction and then we could say our change in Y is our finishing Y point minus our starting Y point all right this is our finishing Y point that's our starting Y point over our finishing x point minus our starting x point and if that confuses you all I'm saying is it's going to be equal to our finishing Y point is negative 6 minus our starting Y point which is 6 over our finishing x point which is 5 minus our starting x point which is negative 1 so this is equal to negative 6 minus 6 is negative 12 5 minus negative 1 that is 6 so negative 12 over 6 the same thing as negative 2 and notice we have a negative slope here negative slope that's because every time we increase X by 1 we go down in the Y direction so this is a downward sloping line it's going from the top left to the bottom right as x increases the Y decreases and that's why we got a negative slope this line over here should have a positive slope let's verify it so I'll use the same points that they use right over there so this is line D slope is equal to rise over run now how much do we rise when we go from that point to that point well let's see we could do it this way we are rising I could just count it out we are rising one two three four five six we are rising six and how much are we running we are running I'll do it in a different color we're running one two three four five six we're running six so our slope is 6 over 6 which is 1 which tells us that every time we move one in the X Direction positive 1 in the X direction we go positive 1 in the Y direction so this is a listed for every X if we go X if we go negative 2 in the X direction we're going to go negative 2 in the Y direction so whatever we do in X we're going to do the same thing in Y in this slope but notice that was pretty easy if we wanted to do it mathematically we could figure out this coordinate right there that we could view as our starting position our starting position is C negative 2 negative 4 negative 2 negative 4 and our finishing position our finishing position is 4 2 4 2 and so our slope change in Y over change in X I'll take this point 2 minus negative 4 2 minus negative 4 over 4 minus negative 2 4 minus negative 2 2 minus negative 4 is 6 and remember that was just this distance right there and then 4 minus negative 2 that's also 6 that's that distance right there and we get a slope of 1 let's do another one let's do another couple and these are interesting let's do the slide II right here so change in Y over change in X so our change in Y when we go from this point to this point we'll just I'll just count it out it's one two three four five six seven eight it's eight or you could even take this y-coordinate to this y-coordinate two minus negative six will give you that distance eight and then what's the change in Y well the Y value here is I'm sorry what's the change in X the x value here is for the x value there is four X does not change so it's eight over zero well we don't know we eight over zero is undefined so in this situation the slope is undefined when you have a vertical line you say your slope is undefined because you're dividing by zero but that tells you that you're dealing probably with a vertical line now finally let's just do this one this seems like a pretty straight up vanilla slope problem right there you have that point right there which is the point three one so this is line F you have the point 3 comma 1 and then over here you have the point negative 6 comma negative 2 so our slope would be equal to change in Y I'll take I'll take I'll take this as our ending point just so you can go in different directions so our change in Y so now we're going to go down in that direction so it's negative 2 minus 1 that's what this distance is right here negative 2 minus 1 which is equal negative 3 notice we went down 3 and then what is going to be our change in X well we're going to go back that amount what is that amount well that is going to be C we start it's going to be negative 6 that's our end point minus 3 negative 6 minus 3 that gives us that distance which is negative 9 so for every time we go back 9 we're going to go down 3 if we go back 9 we're going to go down 3 which is the same thing as if we go forward 9 we're going to go up 3 all equivalent and see these cancel out and you get a slope of 1/3 positive 1/3 it's an upward sloping line every time we run 3 every time we run 3 we rise 1 every time we run 3 we rise 1 anyway hopefully that was a good review of slope for you