Intro to direct & inverse variation

I want to talk a little bit about direct and inverse variation so I'll do direct variation on the left over here and I'll do inverse variation or two variables that vary vary inversely on the right hand side over here so a very simple definition for two variables that vary directly would be something like this why it varies directly with X if Y is equal to some constant with X so we could rewrite this in kind of English as Y varies directly directly with with X and if this constant seems strange to you this just remember this could be literally any constant number so let me give you a bunch of examples that would also be or I guess particular examples of Y varying indirectly with X you could have Y is equal to X because in this situation the constant is 1 we didn't even write it we could write Y is equal to 1 X then K is 1 we could write Y is equal to 2 X we could write Y is equal to 1/2 X we could write Y is equal to negative 2 X we are still varying directly we could have Y is equal to negative 1/2 X we could have Y is equal to pi times X we could have Y is equal to negative pi times X I think I don't want to beat a dead horse now I think you get the point any constant times X we are varying directly and to understand this maybe a little bit more tangibly let's think about what happens and let's pick one of these scenarios let's well I'll take a positive version and a negative version just because it might not be completely intuitive so let's take the version of Y is equal to 2x and let's explore why we say they vary directly with each other so let's pick a couple of values for X and a couple of out and see what the resulting Y value would have to be so if X is equal to 1 then Y is 2 times 1 or is two if X is equal to 2 then Y is 2 times 2 which is going to be equal to 4 so when we doubled X when we went from 1 so we doubled X the same thing happened to Y we doubled we doubled Y so that's what it means when something varies directly if we scale X up by a certain amount we're going to scale up Y by the same amount if we scale down Y by X by some amount we would scale down Y by the same amount and just to show you it works with all of these let's try the situation with y is equal to negative 2x I'll do it in magenta Y is equal to negative well let me do it in a new example that I have even richer let's try Y is equal to negative negative 3x so once again let me do my X and my Y when X is equal to 1 Y is equal to negative 3 times 1 which is negative 3 when X is equal to 2 y so negative 3 times 2 is negative 6 so 1 notice we multiplied so if we scaled let me do that in that same green color if we scale off X by 2 it's a different green color but it serves the purpose we're also scaling up Y by 2 to go from 1 & 2 you multiply it by 2 to go from negative 3 to negative 6 you're also multiplying by 2 so we grew by the same scaling factor and if you wanted to go the other way let's say if we went to let's try a I don't know let's go to 1 X is 1/3 if X is 1/3 then Y is going to be negative 3 times 1/3 is negative 1 so notice to go from 1 to 1/3 we divide by 3 to go from negative 3 to negative 1 we also divide by 3 we also scale down by a factor of 3 so however whatever direction you scale X in you're gonna have the same scaling direction is wide that's what it means to vary directly now it's not always so clear sometimes it will be obfuscated so you could write so let's take this example right over here y is equal to negative 3x and I'm saving this real estate for inverse variation in a second you could write it like this or you could algebraically manipulate it you could maybe divide both sides of this equation ix and then you would get Y over X is equal to negative 3 or maybe you divide both sides by X and then you divide both sides by Y so from this you could get so if you divide both sides by Y now you could get 1 over X is equal to negative 3 times 1 over Y these three statements these three equations are all saying the same thing so sometimes the direct variation isn't quite in your face but if you do this this what I did right here with any of these you will get the exact same result or you could just try to manipulate it back to this form over here and there's other ways we could do it we could divide both sides of this equation by negative 3 and then you would get negative 1/3 Y is equal to X and now this is kind of an interesting case here because here this is X varies directly with Y or we could say X is equal to K some K times y and in general that's true if Y varies directly with X that we can also say that X varies varies directly with Y it's not going to be the same constant it's going to be essentially the inverse of that constant but they're still directly varying now with that said so much said about direct variation let's explore inverse variation a little bit inverse variation the general form if we use the same variables and it always doesn't have to be Y and X it could be an A and a B it could be a it could be a M and an N I'm just you know if I said M varies directly with n we would say M is equal to some constant times n now let's do inverse variation so if I did it with y's and x's this would be y is equal to some constant times 1 over X so instead of being at some constant times X at some constant times 1 over X so let me draw you a bunch of examples it could be Y is equal to 1 over X it could be Y is equal to 2 times 1 over X which is clearly the same thing as 2 over X it could be Y is equal to 1/3 times 1 over X which is the same thing as 1 over 3 X it could be Y is equal to negative 2 over X and let's explore this the first variation the same way that we explored the direct variation so let's pick I don't know let's pick y is equal to 2 over X Y is equal to 2 over X and let me do that same table over here so my table and my x values and my Y values if X is 1 X is 1 then Y is 2 if X is 2 then 2 divided by 2 is 1 so if you multiply X by 2 if you scale it up by a factor of 2 what happens to Y this Y gets scaled down by a factor of 2 you're dividing by 2 now notice the difference here whatever however we scaled X will apply by the same amount now if we scale up X by a factor when we have inverse variation we're scaling down Y by that same so it's that that's where the inverse is coming from and we could go the other way if we made X is equal to 1/2 so if we were to scale down if we were to scale down X we're gonna see that it's going to scale up Y because 2 divided by 1/2 is 4 so here we're scaling up Y so they're going to do the opposite things they vary they vary inverse and you could try it with the negative version of it as well so here we're multiplying we're multiplying by 2 and once again it's not always neatly written for you like this it can be rearranged in a bunch of different ways but it could there as it would still be inverse variation as long as they're algebraically equivalent so you could multiply both sides of this equation right here by X and you would get X Y is equal to 2 this is also inverse variation you would get this exact same table over here you could divide both sides of this equation by Y and you could get X is equal to X is equal to 2 over Y which is also the same thing as 2 times 1 over Y so notice Y varies inversely with X and you could just manipulate this algebraically to show that X varies inversely with Y so X so Y varies inversely with X this is the same thing as saying and we just showed it over here with a particular example that varies inversely inversely with Y and there's other things we could take this and divide both sides by two and you would get Y over 2 is equal to 1 over X there's all sorts of crazy things and so in general if you see an expression for that it relates two variables and they say do they vary inversely or directly or maybe neither you could either try to do a table like this if you scale up X by a certain amount and Y gets scaled up by the same amount then it's direct variation if you scale if you scale up X by something on you might want to try it a couple of different times and you scale down Y you do the opposite with Y then it's probably inverse variation the the more like a surefire way of knowing what you're dealing with is to actually algebraically manipulate the equation so it gets back to either this form which would tell you that it's inverse variation or this form which would tell you that it is direct variation