Determine whether the data in the table is an example of direct inverse or joint

variation. Then identify the equation that represents

the relationship. So let’s just think about what direct

inverse or joint variation even means. So if you have direct variation. Direct variation. So if y varied directly with x it

literally means that y is equal to some constant multiple of x, or if you divide

both sides of this by x it means that y over x is equal to k so the ration

between y and x is a constant. And you could go the other way around. You could also say that x is equal to some constant not, not going to be the same

constant times y. Or that x over y is going to be equal to

some other constant. So these aren’t necessarily the same k. All I’m just saying is that it’s a

constant relationship. These are all examples of direct

variation. In dir, or I should say inverse variation

is to some degree the opposite depending on how

you view the opposite. And before I even talk about that, let’s think about the telltale signs of direct

variation. If x increases, y should increase. So if x increases. Let me do that in the same yellow. So the telltale signs of direct variation,

if x increases then y will increase and vice

versa. The other telltale sign is. Is if you increase x by some, by some

factor. So, if you have x going to 3x then y

should also increase by that same factor. And we could see that with some examples. So, I mean, you could pick a K, let’s say

that, let’s say that K was one. So if y is equal to x, if you take, if x

goes from one to three, then y is also going to go

from one to three. So that’s all we’re talking about here. Let me actually, y should actually to three times y, that’s what I’m talking

about. If you triple x, you’re also gonna end up

tripling y. Inverse variation. You have y being equal to some constant

times one over x. So instead of an x here you have a one

over x or if you multiply both sides by x you get x times y

is equal to some constant. And you could switch the x’s and the y’s

around as well for inverse variation. Now what are the tale tale signs? Well if you increase x, if x goes up, then

what happens to y? If x goes up then this becomes a smaller

value cuz it’s one over x so then y will go

down. Then y will go down. And if you take X and if you’re to say

increase it by a factor of three then what’s going to

happen to Y? Well if you increase this by a factor of three, you’re actually going to decrease

this whole value by a factor of one-third, so Y is going to go, so then you’re going to have one-third

of y. So that’s, these are the tell-tale signs

for inverse variation. Now finally they talk about something

called joint variation, and this one you won’t necessarily see in

introductory algebra course. But joint variation deals with more than

one variable. So if I told you, if I told you that area

of a rectangle is equal to the width of a rectangle times the length of rectangle,

this is an example of joint variation. Area is proportionally to two. Is the proportional to two different

quantities? So the main tell tale sign here for joint

variation frankly is you’re gonna be dealing with

more than two variables. Joint, Joint, Joint variation. So when you look at this example, there

only gi, giving us two variables. So you can rule out joint variation just

right from the get go. Now let’s look at the tell tale signs. So as x is increasing, as x goes from one

to two, what is happening to y? Y went from 12 to six. So as x is going up by a factor of two, y is going, is, is going by a factor of

one half. Or y is being multiplied by one half. So as x goes from one to three, it’s being

multiplied by three. Y is being multiplied or I guess could say

is, is multiplied by one-third. So it’s definitely not direct variation. As x increases, y is decreasing. So it’s definitely not direct variation. And then really you can just rule out,

since we rule out the other two, you can probably guess this is

going to be inverse variation. But we can validate it. When X increases, Y is decreasing. When X increases by a certain factor, Y is increasing by one over that factor, which

is actually decreasing. So, if you go from one to three, if X is

being multiplied by three, then Y essentially

becomes one third of its original value. When X is one Y is 12 when X is three Y is

four so we have inverse variation in play. Now they ask us identify the equation that

represents the relationship. Well we know with inverse variation the

product of x and y need to be equal to some constant. So that if we take x times y over here so

lets just multiply lets make another column here

call this the x times y column. One times 12 is 12. Two times six is 12. Three times four is 12. Four times three is 12. So, clearly in every situation, x times y

is, is a constant and it is 12. So, the equation that represents the

relationship, it is, X, Y is equal to 12 and that is clearly an

inverse

1st one!

Thks!

i learnt more from here than school

this is great. I learned about this in Economics before math. at my school you take Econ. int 10th and Alg. 2 in 11th.

This man's voice is beautiful

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