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CCSS.Math: , , ,

slope is defined as your change in the vertical direction and I could use the Greek letter Delta this little triangle here it's a Greek letter Delta it means change in change in the vertical Direction divided by change in the horizontal direction that is the definition of slope or the standard definition for slope and it's a reasonable way for measuring how steep something is so for example if we're looking at the xy-plane here our change in the vertical Direction is going to be a change in the Y variable divided by change in a horizontal Direction is going to be a change in the X variable and so let's see why that is a good definition for slope well I could draw something with a slope of 1 a slope of 1 might look something like let me I could start it over here and let me get my line tool out so slope of 1 as x increases by 1 Y increases by 1 as x increases by 1 Y increases by 1 so slope of 1 is going to look like is going to look like this is going to look like is going to look like this notice as I have a change in X however much my change in X is so for example here my change in X is plus 2 is positive 2 I'm going to have the same change in Y I my change in Y is going to be plus 2 so my change in Y divided by change in X is 2 divided by 2 is 1 so for this line I have a slope slope is equal to 1 but what would a slope of 2 look like well a slope of 2 should be steeper and we could draw that a slope of 2 I could start at that same point actually why don't I start at the same point and we'll see we don't have to actually start at a different point so if I start over let's say here a slope of 2 would look like a slope of 2 would look like for every one that i increase in the x-direction I'm going to increase 2 in the Y direction so it's going to look like it is going to look like that this line right over here you see it if my change in X is 1 change in X is equal to 1 my change in Y my change in Y my change in Y is - so change in Y over change in X is going to be 2 over 1 the slope here is 2 and now hopefully you're appreciating why this definition of slope is a good one the higher the slope the steeper it is the faster it increases the faster we increase in the vertical direction as we increase in the horizontal direction now what would a negative slope be so let's just think about what a line with a negative slope would mean well a negative slope would mean that we could take an example if we have a if we have our change in Y over change in X or say let's say it was equal to a negative 1 that means that if we have a change in X of 1 then in order to get negative 1 here that means that our change in Y would have to be equal to negative 1 so aligned with a negative slope a negative 1 slope would look like let me see if I can draw it it would look like a negative 1 slope would look like would look like this notice as x increases as x increases by a certain amount so our Delta X here is 1 Y decreases by that same amount instead of increasing so now this is what we consider a downward sloping line so change in Y is equal to negative 1 so our change in Y over change in X change in Y over change in X is equal to negative 1 over 1 which is equal to negative 1 so the slope of this line is negative 1 now if you had a slope with negative 2 it would decrease even faster so a line with a slope of negative 2 it could look something like this let me draw it so as x increases by 1 Y would decrease by 2 as x increases by 1 Y would decrease by 2 so it would look something like let me see if I can it would look like it would look like that notice as our x increases by a certain amount our Y decreases decreases by twice as much so this right over here has a slope this has a slope of negative 2 so hopefully this gives you little bit more intuition for what slope represents and how the the number that we used to represent slope how you can use that to visualize how steep a line is a very high positive slope as x increases it's going to Y is going to increase fairly dramatically if you have a negative slope you're actually going to go from you're actually going to go as x increases your Y is actually going to decrease and then the higher slope this deeper in in the the the more you increase is X increases and the more negative the slope the more you decrease as x increases