# What is an intercept?

I have always taught my students that the $y$-intercept of a line is the $y$-coordinate of the point of intersection of a line with the $y$-axis, that is, for the line given by the equation $y=mx+y_0$, the $y$-intercept is $y_0$. I emphasize that that the $y$-intercept is the number $y_0$ and not the point $(0,y_0)$.

But I was quite surprised when I recently looked at the Wikipedia and Wolfram MathWorld entries for $y$-intercept because these define the intercept as a point and not as a number ("the point where a line crosses the y-axis" and "The point at which a curve or function crosses the y-axis").

Further investigation yielded inconsistencies: the Wikipedia entry for "Line (geometry)" states that in the equation $y=mx+b$, "$b$ is the y-intercept of the line"; the Wolfram MathWorld entry for "Line" states that "The line with $y$-intercept $b$ and slope $m$ is given by the slope-intercept form $y=mx+b$.

Is there a trusted source targeted at mathematics educators (from, say, a government agency, an educational institution, or an organization) that defines "intercept" and consistently uses that definition?

• Closely related to yesterday's question on terminology for critical points.and their components (abscissa and ordinate). – Bill Dubuque Jul 25 '18 at 13:57
• Oddly enough, I was quite surprised in when I found several books that insisted that the $y$-intercept is just the value, and not the point. I've always taught my students that intercepts are point on the graph where the line (or other curve) crosses another line or curve. For example, the circle $x^2 + y^2 = 1$ has two $y$-intercepts: $(0,1)$ and $(0,-1)$. – Xander Henderson Jul 25 '18 at 15:46
• Many ME.SE questions seem to hinge on the fact that mathematical terminology exists on a continuum of standardization/variation, and that there is no single sanctioning body for such definitions. This issue itself almost deserves a canonical Q/A to which we could link. – Daniel R. Collins Jul 26 '18 at 0:26
• In french, it's called "ordonnée à l'origine" which means "ordinate at the origin": $f(0)$. – Paracosmiste Jul 26 '18 at 13:13
• If you look at books.google.com for 19th c. sources, you find definitions of an intercept as "distance from the origin", the "line AB" (A = origin, B on the y-axis) and "AB = h" for y = mx+h, the length of the line cut off on an axis from the origin, one which defines intercept as the distance or the segment, and so on. None defined intercept as a point. Spot checks: These definitions seemed to be used in some textbooks published in 1930s. And in 1950s and 60s, intercepts were given by (scalar) numbers. But in 1967, one gives (a,0) as the intercept. Not important? Trivial distinctions? – user1527 Jul 26 '18 at 23:42

This is a case where you might be looking for a distinction that's pretty subtle.

By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered. I'd be ok with a student's answer to "What is the y-intercept?" to be simply the y value, or the $(0,y_0)$ point.

If a teacher prefers one, you can ask

• What is the y value of the y-intercept?

or

• Give the point (coordinate) of the y-intercept.

When I was in high school, one math teacher was fussy about 'negative' vs 'minus'. He insisted that an answer, "-4" should never be pronounced "minus four". He declared "negative is an adjective, minus is a verb." While I suppose this is true, I never found value in correcting a student who is otherwise doing the math correctly. This case may be similar, if they are getting the concept, don't focus on a point (pun intended) that may be a matter of preference.

• As relevant context, might be worth mentioning that it's the point because you can have intercepts besides the one with the y-axis. – Shufflepants Jul 25 '18 at 17:41
• @JoeTaxpayer I had a teacher who was similarly insistent that "-x" as a quantity was "the opposite of x", not "negative x", because we're not constraining x to be negative. It ... makes a point worth keeping in mind but I agree that it doesn't warrant correction :) – hobbs Jul 25 '18 at 23:46
• By the way, I would say "subtract" is a verb. "Minus" can be as much as adjective as "negative", but I am aware thay many follow a different convention. – J W Jul 26 '18 at 3:54
• I don't get where anyone would think "minus" is a verb. It's a preposition, an adjective, even a noun, but definitely not a verb: en.oxforddictionaries.com/definition/minus – rubenvb Jul 26 '18 at 12:07
• @rubenvb: Lots of people and sources really want numbers to be nouns, and operations to be verbs (I think, for perceived symmetry). This is a fairly common thing to see and in need of correction. (The operations are, of course, indeed really prepositions.) – Daniel R. Collins Jul 26 '18 at 14:14

Some time spent with the Google does not seem to turn up very much. The best answer that I can find seems to be from the Common Core Standards, and is ambiguous, at best. On page 69 of the Core Learning Standards for mathematics (as provided by the State of New York; the same line appears in California's document on page 94), it states that students should be able to

[i]nterpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

There are other references to intercepts, but they are ambiguous, at best. This seems to indicate that the Common Core regards intercepts as values, not points.

A quick perusal of the books on my (and my office-mate's) shelves gives an equally ambiguous answer. Most (though not all) of the precalculus texts say that the line given by $y = mx + b$ has the value $b$ as it's $y$-intercept, while most of my calculus texts define the $y$-intercept to be the point $(0,b)$. A notable exception (notable as it is quite widely used) is a copy of the 6th edition of Stewart's Calculus, which gives the $y$-intercept as a value, not a point.

• Thanks for the references. – Joel Reyes Noche Jul 27 '18 at 9:17

This questions reflects the dangers in over formalization of the language used to discuss simple things. A pedantic speaker might distinguish between the y-intercept b and the intercept (0, b) (although intersection point (0, b) might be a better name for the latter), but very little is gained by fussing about such a distinction, particularly at the elementary level.

The example is very different from that some commenters have referenced that treats the distinction between critical point and critical value. The point on the earth where the temperature is lowest and what that temperature is are two very different things in qualitative terms, and so one needs terminology to distinguish them. In the present case, there is very little difference qualitatively, the fussing is about whether one choose to refer to the point or its ordinate as the y-intercept. Both choices are reasonable and neither is consequential. The difficulty arises mostly because of overly rigid teachers who insist that one is right and the other is wrong because I said so or for some similar reason similarly lacking in reason.

In practice I think it's best to adopt a consistent usage, simply because that is easier for students, but not to fuss when individual students, for having read a book on their own, gone to a tutor, or simply not speaking with the same pedantic care as the teacher, use the other terminology, as long as it is clear that they have answered correctly whatever question requires its use.

• "A definition is only as good as what you can do with it," to paraphrase Ravi Vakil. Both definitions of $y$-intercept are equivalent -- they both allow you to do exactly the same things. – Kyle Miller Jul 26 '18 at 6:48

JoeTaxpayer's answer says the distinction is subtle. To me, the distinction is non-existent.

I don't see any benefit in discerning two concepts that are

1. Closely related
2. Have a trivial bijection between the two concepts
3. Do not appear separately in any other context

In this case, the intercept value $y_0$ is trivially bijected to the intercept point $(0, y_0)$, and the large part of mathematical education is teaching students to recognize equivalent concepts and group them together.

It's worth noting that a real number is a point on the real number line (according to, for instance, Stewart's Calculus, Appendix A). The $y$-axis is a copy of the real number line. So, one could take the defensible position that a $y$-intercept is just as much a point $(0,b)$ as it is a point $b$ along the $y$-axis.

For a little pedantic gloss: linear equations and lines in the plane are different concepts, yet they both have a $y$-intercept. Authors disagree on what the $y$-intercept of a linear equation $y=mx+b$ should be: is it $b$ itself? or is it the point where the graph of the equation intersects the $y$-axis? In any case, an equation is not literally a line, so we have to recognize that it's the same terminology for two different things. In the end, the whole $y$-intercept concept is about capturing a connection between geometry (lines) and algebra (linear equations), and since the connection is so strong it might not be worth forcing a distinction, unless it's to make a pedagogical point.

(I did some digging into geometry, and there are the concepts of intercepted line segments and intercepted arcs. The $y$-intercept is related to the intercepted line segment on the $y$-axis, bounded by the $x$-axis and the given line. Strangely, I couldn't find any mention of calling the actual point where two intersecting lines meet the intercept, except for the $x$- and $y$-intercepts.)

It comes down to convention, ultimately, since specifying a unique point $y_0$ associated to $x_0$ is equivalent to specifying the ordered pair $(x_0,y_0)$ on the graph of $f(x)=y.$ That is, since a function is well defined by definition, then there is no ambiguity if one requires that the intercept "$y_0$" be associated to a given input "$x_0$".

$y$ is a value.

The difference between the two would essentially boil down to dimensions of $y$, with y-as-value being uni-dimensional and y-as-point being n-dimensional. Now, even with multiple $x$'s, all information about the intercept (think about difference in the intercepts of two models) is captured in y-as-value. The other dimensions never come into play.

• Can you provide a reference for this assertion? – Xander Henderson Jul 31 '18 at 13:09
• This is my interpretation of explanations from standard textbooks. – skoh Aug 1 '18 at 17:27