Direct and Inverse Variation Understanding direct and inverse variation
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- I want to talk about direct and inverse variations
- I´ll do direct variations on the left over here
- And I´ll do inverse variations or two variables that vary inversly on the right hand side over here
- So a very simple definition for two variables that vary directly
- would be something like this
- Y varies directly with X if y is equal to some constant with x
- we could rewrite this in kind of english
- y varies directly with x
- if this constant seems strange to you
- remember this could be literally any constant number
- so let me give a bunch of examples that would also
- or particular examples of y varies directly with x
- you cold have Y is equal to X
- because of the situation the constant is 1, we didn´t even write it
- you could have write Y is equal to 1X, then K is 1
- we could write Y is equal to 2X
- we could write Y is equal to one half X
- we could write Y is equal to negative 2X
- we are still variating directly
- we could write Y is equal to negative one half X
- we could have Y is equal to π times X
- we could have Y is equal to negative π times X
- You get the point
- Any constant times X, we are varying directly
- And to understand this maybe a little bit more tanginbly
- let´s think about what happens and let´s pick one of these scenarios
- 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 see what the resulting Y value would have to be
- so if X is equal to 1, then Y is two times 1, so it´s 2
- 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, from 1 to 2, so we doubled X, the same thing happened to Y
- We doubled Y, so that´s what it means when something varies directly
- If we scale X up by a certain amount, we are going to scale up Y
- by the same amount
- If we scale down X by a certain 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 ..., I´ll do a new example, let´s try Y is equal to 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, negative 3 times 2 is negative 6
- so notice, we multiplied, so if we scaled, let me do that in the same green colour
- if we scale up X by 2, it´s a different green colour but it serves the purpose
- we are also scaling up Y by 2, to go from 1 to 2, we multiply 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 want to get the other way
- let´s say we try, X is 1/3, if X is 1/3 then Y is going to be negative 3 times 1/3, is negative 1
- so 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 whatever direction you scale X in you are going to have the same scaling direction as Y
- that´s what it means to vary directly
- now it´s not laways so clear
- sometimes it would be ofuscated
- so you could write, so let´s take this example right over here
- Y is equal to negative 3X, (I´m saving this real state for inverse variation in a second)
- you could writethis or you could algebraically manipulate it
- you could maybe divide both sides of this equation by X
- and then you will get Y/X is equal to -3
- or maybe you divide both sides by X
- and then you divide both sides by Y
- so from this you would get, so if you divide both sides by Y now
- you could get 1/X is equal to negative 3 times 1/Y
- these three statements, these three equations are all the same thing
- so sometimes the direct variation isn´t quite in your face
- but if you do this, what I did right here with any of these
- you would get the exact same result
- or you could jus 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 1/3Y is equal to X
- and 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 some K times Y
- and in general that´s true
- if Y varies directly with X, though we can also say that X 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 about direct variation
- let´s explore inverse variation
- inverse variation the general form (if we use the same variables)
- it always doesn´t have to be Y and X, it could be A and a B
- it could be a M and an N
- now to inverse varitaion, so if I did it with Ys and Xs
- this would be Y is equal to some constant (K) times 1/X
- so instead of being some constant times X, it´s some constant times 1/X
- so let me show you a bunch of examples
- it could be Y is equal to 1/X
- it could be Y is equal to 2 times 1/X
- which is clearly the same thing as 2/X
- it could be Y is equal to 1/3 times 1/X
- whis is the same thing as 1/3X
- it could be Y is equal to -2/X
- and let´s explore this, the inverse variation the same way we explain the direct variation
- so let´s pick Y is equal to 2/X
- and let me do that same table over here
- if 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
- you scale it up by a factor of 2, what happens to Y?
- Y gets scaled down by a factor of 2
- you divide it by 2 now
- notice the difference?
- here however we scale X, we sacled up Y by the same amount
- now if we scale up X by a factor
- when we have inverse variation, we are scaling down Y by the same
- so that´s were the inverse is coming from
- and we could go the other way
- id we made X is equal to 1/2
- so if we were to scale down X, we are going to see that this is going to scale up Y
- ´cause 2 divided by 1/2 is 4
- so here we are scaling up Y
- so they are going to do the opposite things
- they vary inverse, and you could try it with the negative way of it as well
- so here we are multiplying by 2
- and once again it is not always neatly written for you like this
- it could be rearranged in a bunch different ways
- but it will still be inverse variation as long as they are algebraically equivalent
- so you could multiply both side of this equation right here by X
- and you would get XY 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 X is equal to 2/Y
- which is also the same thing as 2 times 1/Y
- so notice Y varies inversely with X
- and you could just manipulate this algebraically to show that X varies inversely with Y
- so Y varies inversely with X
- this is the same thing (as seen in this example) as saying that X varies inveresely with Y
- and there is other things
- we could take this and divide both sides by 2
- and you would get Y/2 is equal to 1/X
- there´s all sort of crazy things
- so in general if you see an expression that relates two variables
- and they say do they vary inversely or directly or maybe neither
- you could either try to make a table like this
- if you scale up X by certain amount and Y gets scaled up by the same amount
- that is direct variation
- if you scale up X, you may want to try it a couple of different times
- and you scale down Y, do the opposite with Y, then it is probably inverse variation
- a sure way to know what you´re dealing with
- it´s to actually algebraically manipulate the equation so it gets back to either this form
- that tells you it´s inverse variation
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