Question

Dear Mitch,

I'm in seventh grade advanced math, and we're doing different kinds of graphs and "functions", but I don't understand something about the graphs we're doing. When you have an x and a y, like there usually is in the graphs, why would it matter in the real world which thing you pick for your x's and which you pick for your y's? (I know it has something to do with one of them being something called the 'independent variable' and one of them being the 'dependent variable', but I'm not even sure how you can tell which is which.) I usually understand things in math, but I really am hoping you can explain it in a way that makes more sense than it's making to me right now.

Sincerely,

Not Feeling Independent

Answer

Dear Soon Independent Again,

A lot of people get confused by the same things you're wondering about, and many of them want to ask the questions you asked, but a lot of the times they are too confused and worried about 'getting it' that they don't even have their questions as clear in their heads as you do. So, the good news is you are already halfway toward 'getting' everything you're asking! (And I would even say that the half you've made it through – figuring out exactly what your questions are -- is the more difficult half!).

First of all, I agree with you that it does seem that it shouldn't matter which of the two variables (the thing that changes or the thing that at least *can* change) you decide to represent on the horizontal axis, and which you represent for the vertical axis. For example, if you are graphing how many minutes people brush their teeth on average and the average number of cavities they have to get filled at the dentist each visit, it would seem that you could label the horizontal part of the graph EITHER: "NUMBER OF MINUTES OF BRUSHING TEETH PER DAY" **or** "NUMBER OF CAVITIES THAT REQUIRED FILLING THAT YEAR".

In either case, I think a person would be able to figure out what the graph meant, and my guess would be that the more time a person spends brushing his/her teeth, the fewer cavities he/she is likely to have that require filling at the dentist's office. BUT, the relationship between the two would not (and could not) be completely perfect, because even a quick think leads one to come up with likely exceptions that would prevent the graph from indicating that its point was true 100% of the time. (Very few things in the world are true 100% of the time, though many things come close. One of the reasons we graph information is to see how close the relationship is between the alteration of a particular variable (ingredient) and the change in the outcome resulting from that alteration.)

So, if you think about the fact that most of math was developed as a language to describe what happens in nature to assist us in at least *trying to* make predictions that will help us survive and then gain more knowledge to survive even better as time goes on, then it should come as no surprise to learn that graphs are one of the tools we have to do this. SO...

Let's say we are graphing people's age and their height. A medicine company might want to do this because if a certain treatment or dosage is based upon a person's height or age or a combination of the two, the company has to be able to look at the results of how much treatment works best for different people so they can make recommendations on the side of the bottle or in medical textbooks that doctors may refer to when making difficult decisions. SO...

Most children would guess that the relationship between age and height goes like this: As you get older, you grow taller. And that certainly is true much of the time. Most children look forward to their birthdays because they know just from looking around the playground that as people get older they usually get bigger and stronger, which, of course, are two things that a lot of young people REALLY WANT. BUT...

It is also true that when height and age change, one causes the other. The change in age CAUSES the change in height. So, even though a child is likely to get a little taller in a year, it is not true that if that child goes on a special super-healthy diet and exercise program and gets taller super-fast, the diet has made him grow older; no, his birthday will not come sooner. If you put that child on a machine that stretches him, and the child becomes taller *that* way, that child also does not immediately become older.

And finally, many people find that as they get *much* older, say between 70-years-old and 95-years-old, they become a tiny bit shorter. (And this is probably a good thing, because longer bones can break more easily when a person falls, and recovering from broken bones takes longer for a person whose body is slowing down a little.) SO...

When you understand that age is the independent variable (the cause), and height is the dependent variable (the result) you can graph it in a way that shows this cause/effect relationship. OTHERWISE, the graph, instead of helping people understand something, can end up being very confusing: Since a person can be the SAME HEIGHT at TWO DIFFERENT AGES, but NOT two DIFFERENT HEIGHTS at the SAME AGE, It is important for the graph to be drawn in the way that makes the information most helpful for making predictions.

Back to the teeth and cavities: If the independent variable and the dependent variable are mixed up, it could lead to a graph that makes it seem like the less cavities a person has filled, the more excited they get at the dentist's office to make them run home and brush more and more and more. Likewise, as a person is told they have more cavities, they usually don't become so angry at their teeth that they stop brushing them!

No, it's usually the care of the person's teeth that results in the dentist's report. Usually. (Of course, sometimes just one slippery baseball bat flying out of a batter's hands and into an unfortunate child's mouth can put a little dent in the graph's smooth flow – especially if the bat hit teeth that the child absolutely loved to brush and brush and brush...

SO, now that you've read through all that, here's the FAST ANSWER and a TRICK to help:** A graph shows a function if a change in one variable produces only one result each time. In other words, addition would not be a function if 2 + 2 = 4, 5, and 6.**

**The TRICK to test a graph for this is called the VERTICAL LINE TEST. ALL you do is you hold your pencil vertically and slide it along the graph. If the graph ever crosses the pencil more than once at the same time (or the pencil crosses the graph more than once at the same time), the graph is NOT showing a "FUNCTION".**

Hope that helps!

Sincerely,

Mitch