We can distinguish 3 groups of equations depending on whether they have a y-intercept only, an x-intercept only, or neither. The first group (y-intercept only) can have almost any type of equation, including linear equations. A good easy example is y = 3 (or any other constant value of y except for 0) since this is a line parallel to the x-axis and will, thus, never cross or intercept it. Please don't try to calculate these types of intercepts on this slope intercept form calculator as these types of equations can potentially break the Internet.

The second and third groups of equations are a bit more tricky to imagine and to understand them well, we need to introduce the concept of an asymptote. An asymptote is a line (that can be expressed as a linear equation) to which the function or curve we are talking about gets closer and closer but never actually crosses or touches that line.

The definition might not seem totally clear, but if we look at an example equation, we will have fewer problems with understanding it. Let's take the equation y = 1/x. If we try to find the y-intercept by substituting x = 0, we arrive at what is called a mathematically undefined expression since it makes no sense to divide by 0.

If we take values closer and closer to 0 (something like 0.1, then 0.001, 0.000001...), we can see that the value of y increases very rapidly. So around the point x = 0, we know that y would have a massive value, but because of how math works, it does not have a defined value for that exact point. Sometimes people may say 1/0 = ∞, but the reality is that infinity is not a number but a concept.

In this case, the linear equation x = 0 represents the asymptote of the function y = 1/x, which means that y = 1/x will never intercept that line and, thus, will not have a y-intercept. In general, any time that a function has an asymptote that lies on one of the axes, it will be missing at least one of the intercepting points.

In fact, the example we have shown you (y = 1/x) also has an asymptote for y = 0, i.e., the x-axis. For the same reason as before, y = 0 is never achievable by the formula because it would require x = ∞, and as we said before, it is impossible to achieve that since infinity is a concept and not a number.

Before we move to our next topic, it is important to note that we have made extreme over-simplifications when talking about infinity, but we feel it is a good and fast approach for those that are not used to the concept of working with infinity in math. We recommend that you learn more about the proper ways of infinity, starting with the undefined expressions in math.