# Question Video: Evaluating the End Behavior of Rational Functions Mathematics • 10th Grade

Consider the graph of the function π¦ = 1/π₯. By looking at the graph and substituting a few successively larger values of π₯ into the function, what is the end behavior of the graph as π₯ increases along the positive π₯-axis? (a) The value of π¦ approaches infinity as π₯ increases. (b) The value of π¦ approaches negative infinity as π₯ increases. (c) The value of π¦ approaches zero as π₯ increases.

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### Video Transcript

Consider the graph of the function π¦ equals one over π₯. This question has three different parts. Letβs start with the first part. By looking at the graph and substituting a few successively larger values of π₯ into the function, what is the end behavior of the graph as π₯ increases along the positive π₯-axis? (a) The value of π¦ approaches β as π₯ increases. (b) The value of π¦ approaches negative β as π₯ increases. (c) The value of π¦ approaches zero as π₯ increases.

This part is specifically asking us to look at the positive π₯-axis. That would be this space. We need to know what is happening to the π¦-values as we move from left to right along the positive π₯-axis. We noticed that the π¦-values are decreasing. And so, we could say that the π¦-values are not approaching β. And that means we need to answer the question, are these values approaching negative β or zero?

To answer this, we can substitute some larger values in for π₯. We know that π¦ equals one over π₯. If we plug in 10 for the value of π₯, then π¦ will equal one-tenth. If we write that as a decimal, that is 0.1. If we plug in 100 for the value of π₯, π¦ equals one over 100. If we write that as a decimal, it is 0.01. It looks like these values are getting closer and closer to zero. And this makes sense because π₯ must be a positive value since weβre only considering the positive values of π₯. And if π₯ is positive, then one over π₯ could not be negative. As π₯ increases, the fraction one out of π₯ gets closer and closer to zero, which means option (c) is correct. The value of π¦ approaches zero as π₯ increases.

Part two says, similarly, what is the end behavior of the graph as π₯ decreases? (a) The value of π¦ approaches zero. (b) The value of π¦ approaches β. Or (c) the value of π¦ approaches negative β.

This time weβll start at π₯ equals zero and move to the left. Weβll be considering the negative values of π₯. As we move further and further to the left, we get more and more negative values. Those are the smallest values π₯ can be. And this time, as we move to the left, we see that the π¦-value is increasing. Itβs coming up. So, letβs think about our function π¦ equals one over π₯. If we plug in negative 10 for π₯, π¦ equals negative one-tenth. If we write that as a decimal, it is negative 0.1. And if we make our π₯-value much smaller, we could plug in negative 100. Negative 100 is less than negative 10 because negative 100 is to the left of negative 10 on a number line. And if we write negative one over 100 as a decimal, we get negative 0.01.

Looking at these two values, itβs a little bit harder to see whatβs happening. But we can eliminate one option. We can eliminate option (b) that says the value of π¦ approaches β. If our function is one over π₯ and weβre plugging in negative values for π₯, the outcome is going to be negative and, therefore, will not be approaching positive β. If we look at our graph, we can answer this question. This line is getting closer and closer to the π₯-axis, and the π₯-axis is the place where π¦ equals zero. Negative one hundredth is smaller than negative one-tenth. Negative one hundredth is closer to zero. And so, we can say that the end behavior as π₯ decreases is the value of π¦ will approach zero.

Part c: finally, by interpreting the graph, what is happening to the function when the value of π₯ approaches zero?

We have to remember that π₯ can approach zero from the right side or from the left side. This is important because, sometimes, the behavior as π₯ approaches zero from each side is different. As π₯ approaches zero from the positive side, the graph is going up. As π₯ approaches zero from the negative side, the graph is going down. The graph going up, we can say, is approaching positive β. And the graph going down is then approaching negative β. And so, we can say the value of π¦ approaches negative β when π₯ gets closer to zero from the negative direction, and that π¦ approaches positive β when π₯ gets closer to zero from the positive direction.

And so, weβve seen four different behaviors. As π₯ decreases, the π¦-values approach zero. As π₯ increases, the π¦-values approach zero. As π₯ approaches zero from the negative side, our π¦βs go toward negative β. And as π₯ approaches zero from the positive side, our π¦-values go toward positive β.