A Leap Day 2016 Mathematics A To Z: X-Intercept

Oh, x- and y-, why are you so poor in mathematics terms? I brave my way.


I did not get much out of my eighth-grade, pre-algebra, class. I didn’t connect with the teacher at all. There were a few little bits to get through my disinterest. One came in graphing. Not graph theory, of course, but the graphing we do in middle school and high school. That’s where we find points on the plane with coordinates that make some expression true. Two major terms kept coming up in drawing curves of lines. They’re the x-intercept and the y-intercept. They had this lovely, faintly technical, faintly science-y sound. I think the teacher emphasized a few times they were “intercepts”, not “intersects”. But it’s hard to explain to an eighth-grader why this is an important difference to make. I’m not sure I could explain it to myself.

An x-intercept is a point where the plot of a curve and the x-axis meet. So we’re assuming this is a Cartesian coordinate system, the kind marked off with a pair of lines meeting at right angles. It’s usually two-dimensional, sometimes three-dimensional. I don’t know anyone who’s worried about the x-intercept for a four-dimensional space. Even higher dimensions are right out. The thing that confused me the most, when learning this, is a small one. The x-axis is points that have a y-coordinate of zero. Not an x-coordinate of zero. So in a two-dimensional space it makes sense to describe the x-intercept as a single value. That’ll be the x-coordinate, and the point with the x-coordinate of that and the y-coordinate of zero is the intercept.

If you have an expression and you want to find an x-intercept, you need to find values of x which make the expression equal to zero. We get the idea from studying lines. There are a couple of typical representations of lines. They almost always use x for the horizontal coordinate, and y for the vertical coordinate. The names are only different if the author is making a point about the arbitrariness of variable names. Sigh at such an author and move on. An x-intercept has a y-coordinate of zero, so, set any appearance of ‘y’ in the expression equal to zero and find out what value or values of x make this true. If the expression is an equation for a line there’ll be just the one point, unless the line is horizontal. (If the line is horizontal, then either every point on the x-axis is an intercept, or else none of them are. The line is either “y equals zero”, or it is “y equals something other than zero”. )

There’s also a y-intercept. It is exactly what you’d imagine once you know that. It’s usually easier to find what the y-intercept is. The equation describing a curve is typically written in the form “y = f(x)”. That is, y is by itself on one side, and some complicated expression involving x’s is on the other. Working out what y is for a given x is straightforward. Working out what x is for a given y is … not hard, for a line. For more complicated shapes it can be difficult. There might not be a unique answer. That’s all right. There may be several x-intercepts.

There are a couple names for the x-intercepts. The one that turns up most often away from the pre-algebra and high school algebra study of lines is a “zero”. It’s one of those bits in which mathematicians seem to be trying to make it hard for students. A “zero” of the function f(x) is generally not what you get when you evaluate it for x equalling zero. Sorry about that. It’s the values of x for which f(x) equals zero. We also call them “roots”.

OK, but who cares?

Well, if you want to understand the shape of a curve, the way a function looks, it helps to plot it. Today, yeah, pull up Mathematica or Matlab or Octave or some other program and you get your plot. Fair enough. If you don’t have a computer that can plot like that, the way I did in middle school, you have to do it by hand. And then the intercepts are clues to how to sketch the function. They are, relatively, easy points which you can find, and which you know must be on the curve. We may form a very rough sketch of the curve. But that rough picture may be better than having nothing.

And we can learn about the behavior of functions even without plotting, or sketching a plot. Intercepts of expressions, or of parts of expressions, are points where the value might change from positive to negative. If the denominator of a part of the expression has an x-intercept, this could be a point where the function’s value is undefined. It may be a discontinuity in the function. The function’s values might jump wildly between one side and another. These are often the important things about understanding functions. Where are they positive? Where are they negative? Where are they continuous? Where are they not?

These are things we often want to know about functions. And we learn many of them by looking for the intercepts, x- and y-.