Question

John was graphing a function and noticed that at certain points, the graph reaches invisible lines the graph will never cross. Explain to John what the two types of invisible lines are and how to predict them. You may create your own example to aid in your reasoning. Use complete sentences.

Answer 1

The invisible lines are call asymptotes, which are certain values that the graph will never reach or can't go through due to constraints.

For example, a vertical asymptote is noticed when you can't plug in a certain x-value into the denominator of a function. This is because you are NOT allowed to divide by zero at any time.

\[\frac{ x+4 }{ x-3 }\] The above equation demonstrates this since you cannot plug in x=3 into this function. Doing so will result in a 0 in the denominator, which is not allowed. As a result, there is a vertical asymptote at x=3.

Answer 2

Another type of asymptote is a horizontal asymptote. This is seen when there is a certain value that y will only reach and never approach.

\[\frac{ 1 }{ x }\]
The above equation is a classic example of this (and a vertical asymptote!) since the larger that you make x, the smaller the result will be. This makes it seem that the graph is going to be 0 at some point since it is quickly approaching the x-axis, but will never reach it. Therefore, y=0 is a horizontal asymptote of this function.