In this lesson, our instructor Vincent Selhorst-Jones teaches Variation Direct and Inverse. Youll learn the definitions of direct, inverse, joint, and combined variation. Vincent also uses real world examples like groceries and gravity to explain the concepts. The lecture concludes with four practice examples to help reinforce the lesson.
Variation is a way of talking about how different things relate to each other. Depending on the type of variation, we'll know what form the relationship takes. However, we should note that variation isn't connected very closely to functions.
While you can describe it in the language of functions, it's easier to talk about variation with equations, so that's what we'll do.
Direct variation says that two things are directly related to each other. If one goes up, the other goes up. If one goes down, the other goes down. They (usually) go at different rates, but the same "direction".
There are many different ways to say two things (let's call them x and y) are in direct variation:
x and y vary directly;
y varies directly as x;
x and y are directly proportional;
y is directly proportional to x.
In any case, all these phrases mean the same thing mathematically:
y = k ·x,
where k is a constant. It is called the proportionality constant or the constant of variation. It is the rate at which the two things are connected.
Inverse variation is the opposite of direct variation. It says that two things are inversely related to each other. If one goes up, the other goes down. If one goes down, the other goes up. They (usually) go at different rates, but opposite "directions".
[Caution: inverse variation is not related to inverse functions. It's based on the idea of multiplicative inverses, like 3 and [1/3].]
Like direct variation, there are many alternative ways to say that x and y are in inverse variation:
x and y vary inversely;
y varies inversely as x;
x and y are inversely proportional;
y is inversely proportional to x.
Not only that, but the relationship is sometimes called a reciprocal proportion or (confusingly) indirect variation. Still, they all mean the same thing mathematically:
where k is a constant. It serves the same purpose as it does for direct variation: it gives the rate at which the things are connected.
Joint variation is a variation where multiple direct variations are happening at the same time. We could say:
z varies jointly as x and y;
z is jointly proportional to x and y.
Both of these would mean
z = k·x ·y,
where k is once again a proportionality constant.
We can also combine direct and inverse variation if we have these sorts of relationships going on simultaneously. Pay attention to what kinds of variation each variable gives, then put them all together. Notice that no matter how many variations are
put together, you only need a single constant k.
Variation Direct and Inverse
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.