All lines with a slope have a y-intercept?

I must be missing something elementary (likely, since I’m working on an elementary text). I’m reading a sentence that says that the “y-intercept is a shared characteristic of all lines with slope.” But what about, say, a parabola that is asymptotic to the y-axis (e.g., shifted over to the right)? It never reaches the y-axis, so there is no intercept. Is it because at the limit/asymptote the slope is undefined? But other parts of the line do have a determinate slope, and since it’s a continuous function that part with a slope does not have a y-intercept. Would it be better to say that the “y-intercept is a shared characteristic of all linear functions with a defined slope”?

The word “line” can be generally assumed to mean linear function. If the discussion included parabola or other non-linear functions, they probably would have said “curve” or similar.

So if I was being as pedantic/technical as possible, I wouldn’t make that assumption? Or is that crossing the line of wayyyy too pedantic? I have to remain very basic and approachable but remain beyond reproach from a math-nazi.

While we’re at it with the pedantry, I’ll point out that parabolae don’t have asymptotes. You may be thinking of a hyperbola.

Oops… thanks!

(Would it be hyperbolic to say I love the Dope?)

I’d call it way too pedantic. As Rhubarb says, “line” means a line, not some other kind of curve. In case there is any doubt, you can tell this is what they’re getting at because of the phrasing “all lines with slope”; i.e., they consider slope to be a characteristic of a “line” in itself, and not just specific points upon it (as would be necessary were we discussing curves in general, whose instantaneous slopes vary from point to point). Furthermore, this wording also properly rules out vertical lines (whose slope is not a real number), as those lack y-intercepts (which is to say, they either intersect the y-axis nowhere or everywhere, but, in either case, not at any one point).


No, you’d just be going off on a tangent.

I guess you have to draw a line somewhere.

Focus, people.

I’m looking for some way to connect all these points.

I ass assume, when you say “y-axis” you mean the axis parallel to a line with a slope of zero. In other words, what most people call the “x-axis”. I’ve seen the z and y axis get mixed and matched in a 3d graph, but I’ve never seen that axis called the y-axis.

Or am I having a major senior moment? (I know, I’m not even forty yet. I’m just using hyperbole.)

It is the y-axis. The point is that any non-vertical (straight) line intersects the vertical y-axis at some point. And vertical lines have an undefined or infinite slope, so therefore all lines with a (non-infinite) slope intersect the y-axis.

A slope of zero is defined, so horizontal lines do have a slope, and the statement “All lines with a slope intersect the x-axis” is not true.

There are three cases to consider:
(1) lines with a slope of 0, i.e., parallel to the x axis. They will have an equation like y=c, and will cross the y axis at the point (0,c).
(2) lines with a non-zero slope. If the slope is s, they will have an equation like y=sx+c, and again will cross the y axis at the point (0,c).
(3) lines without a slope because they are parallel to the y axis, and having an equation of the form x=c.

I’m not sure why you think the question is putting them the other way round.

The most common convention is

y = mx + b

where m is the slope and b is the y intercept. No difference in meaning, and not the only “right” way, just the predominant convention.

I see, my problem was equating a slope of zero to “no slope”. I would have said “undefined slope”. I also might have said infinite slope even though that may not be rigorously correct.

This is an interesting point. I do not know if the phrase “having no slope” is used rigorously in mathematics. In real life, if you are standing on a perfectly level road, you would not say, “I am on a slope, it’s just that the slope is zero.” :slight_smile: You would say the road has no slope.

No, just elliptical. :wink:

My father the math professor would give me a very big lecture whenever I tried to substitute words like “no” or “none” or “nothing” with “zero.” In mathematics, they are not equivalent.

But a flat road does have a slope of zero, because you have no “rise” (increase in y) but you do have a defined “run” (increase in x) (assuming you’re at the origin, going in increasing x direction, etc).

Basically a flat road has a slope of m = (0-0)/(X2-X1) = 0 for all X2 != X1. You can say “no slope”, but it is definitely defined as a slope of zero. This equation is y = c.

But, if you’re not moving along the road, but being levitated by an alien spaceship (what? It could happen!) then you’ve got a “rise” but no “run”, and your slope becomes

m = (Y2-Y1)/(0-0) = undefined, because division by zero isn’t defined. This is an x=c equation.

And of course, if you’re climbing a linear hill while walking the road, you have a rise and a run, and your slope is m = (Y2-Y1)/(X2-X1), for an equation that is y = mx+b. When x = 0, y = b, which is your intercept.