I’ve been teaching and tutoring the slope intercept form of the equation of a line for 25 years. That whole time I’ve wondered, “Why is the y-intercept called b?”
Just today I discovered the answer!
We teach it backwards.
As is typical, we teach things in reverse of how we create them. The standard (or general) form of a linear function is
f(x) = ax + b
But this crazy f(x) notation isn’t really needed when you first learn to graph. We use y instead. So we have
y = ax + b
The nifty thing about this equation is that the number next to the x (in this case a) is the slope – or the measurement of how much the line tilts.
The fancy letter for the slope is m. So we change the basic equation to include the slope notation…
y = mx + b
Turns out that the b here is also exactly where the line smacks into the y-axis – AKA the y-intercept. But there’s no fancy letter for that, so we just leave it as b.
And we present that first!
We tell early students of algebra that the slope intercept form of the equation of a line is
y = mx + b
where m = slope and b = y-intercept. But we don’t bother to tell them why we’re using such crazy letters!
But now you have it. That’s the answer to the question, “Why is the y-intercept called b?”
(If only we could figure out why they called the slope m!)
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Hmm… I’ve heard different reasoning on this. I’ve always thought b was for the y-intercept because a was the variable used for the x-intercept number. I’ve seen “a” being used in various math texts particular in the context of graphing linear equations after finding the two intercepts. “Intercept form” section on http://en.wikipedia.org/wiki/Linear_equation
describes this in brief. Now I’m not sure which is more accurate; your explanation makes sense too 🙂
And it could possibly be merely the preference of the person writing a textbook. And if that textbook author wrote it long ago, and the textbook was adopted for significant time, it becomes a standard just by means of tradition.
We may never really know for sure.
Perhaps the key take away is to explain it in some manner so that your student can connect with it.
Thanks for stopping by, 314girl!
It gets even more confusing to students when you try and teach the standard form of a line:
Ax + By = C. Why did it switch to capital letters?
The constant term which was the y intercept in slope/intercept form is not the y intercept any more. It seems like we could do this in a more logical fashion.
The standard form of the equation of a line has always gotten on my nerves. I’ve never identified why, but maybe this is it.
Each new mathematician or math teacher comes along and thinks, “My students would understand it better this way.” So they write the book that way.
Before we know it we’ve got various and sundry ways of explaining the same thing all using different notation.
And we wonder why students say, “You just created this to make me confused.”
Thanks for pointing this out, Barbara.
I was actually in the third year of my college teaching job before finding out that m is the first letter of the French verb monter, “to climb”, hence its (French) connection to the idea of slope.
Only took 25 years beyond that to get the explanation of b. Thanks.
I discovered your site about six hours ago and am greatly enjoying it.
Thanks, Guy! I’m glad you’re enjoying it.
Great explanation. As for “m”, it stands for “montagne” which is translated to mountain. The first equations using slope denoted it as “montagne” because slope was related to mountains and their increase/decrease over distances. So, we replaced “a*x” with “m*x” to help students denote rate of change or slope/increase! Hope this helps!
Whoa! Thanks, Nathaniel!
I’ve been showing it as a mountain for years, never knowing that it really was from that.
Thanks a bunch for jumping in and sharing!
I’m sorry. I’m a student and the slope equation is such a pain for me that I can’t understand anything in coordinate geometry for my GMAT.
I just don’t understand why the the Y in the equation is!
Why is the B the Y intercept and not Y? And when we are trying to find Y=mx+b, what exactly are we looking for? What do we find? Y? What is why? and why is b the Y intercept and Y is something that has an equation but no name? It’s just so frustrating!
Brian, I’m sorry you’re frustrated.
The y represents numbers that can vary depending on what is used for x. The b is a single specific number.
The rub here is when letters are used as variables, compared to when letters are used as constants.
It’s sticky to keep track without a lot of practice.