Video: Evaluating the End Behavior of Rational Functions

Video Transcript

Consider the graph of the function 𝑦 equals one over 𝑥. This question has three different parts. Let’s start with the first part. By looking at the graph and substituting a few successively larger values of 𝑥 into the function, what is the end behavior of the graph as 𝑥 increases along the positive 𝑥-axis? (a) The value of 𝑦 approaches ∞ as 𝑥 increases. (b) The value of 𝑦 approaches negative ∞ as 𝑥 increases. (c) The value of 𝑦 approaches zero as 𝑥 increases.

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

To answer this, we can substitute some larger values in for 𝑥. We know that 𝑦 equals one over 𝑥. If we plug in 10 for the value of 𝑥, then 𝑦 will equal one-tenth. If we write that as a decimal, that is 0.1. If we plug in 100 for the value of 𝑥, 𝑦 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 𝑥 must be a positive value since we’re only considering the positive values of 𝑥. And if 𝑥 is positive, then one over 𝑥 could not be negative. As 𝑥 increases, the fraction one out of 𝑥 gets closer and closer to zero, which means option (c) is correct. The value of 𝑦 approaches zero as 𝑥 increases.

Part two says, similarly, what is the end behavior of the graph as 𝑥 decreases? (a) The value of 𝑦 approaches zero. (b) The value of 𝑦 approaches ∞. Or (c) the value of 𝑦 approaches negative ∞.

This time we’ll start at 𝑥 equals zero and move to the left. We’ll be considering the negative values of 𝑥. As we move further and further to the left, we get more and more negative values. Those are the smallest values 𝑥 can be. And this time, as we move to the left, we see that the 𝑦-value is increasing. It’s coming up. So, let’s think about our function 𝑦 equals one over 𝑥. If we plug in negative 10 for 𝑥, 𝑦 equals negative one-tenth. If we write that as a decimal, it is negative 0.1. And if we make our 𝑥-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, it’s a little bit harder to see what’s happening. But we can eliminate one option. We can eliminate option (b) that says the value of 𝑦 approaches ∞. If our function is one over 𝑥 and we’re plugging in negative values for 𝑥, the outcome is going to be negative and, therefore, will not be approaching positive ∞. If we look at our graph, we can answer this question. This line is getting closer and closer to the 𝑥-axis, and the 𝑥-axis is the place where 𝑦 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 𝑥 decreases is the value of 𝑦 will approach zero.

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

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

And so, we’ve seen four different behaviors. As 𝑥 decreases, the 𝑦-values approach zero. As 𝑥 increases, the 𝑦-values approach zero. As 𝑥 approaches zero from the negative side, our 𝑦’s go toward negative ∞. And as 𝑥 approaches zero from the positive side, our 𝑦-values go toward positive ∞.