Video: Evaluating the End Behavior of Rational Functions

Consider the graph of the function 𝑦 = 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 𝑦 approaches infinity as π‘₯ increases. (b) The value of 𝑦 approaches negative infinity as π‘₯ increases. (c) The value of 𝑦 approaches zero as π‘₯ increases.

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Video Transcript

Consider the graph of the function 𝑦 equals one over π‘₯. 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 π‘₯ into the function, what is the end behavior of the graph as π‘₯ increases along the positive π‘₯-axis? (a) The value of 𝑦 approaches ∞ as π‘₯ increases. (b) The value of 𝑦 approaches negative ∞ as π‘₯ increases. (c) The value of 𝑦 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 𝑦-values as we move from left to right along the positive π‘₯-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 π‘₯. We know that 𝑦 equals one over π‘₯. If we plug in 10 for the value of π‘₯, 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 π‘₯, 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 π‘₯ must be a positive value since we’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 𝑦 approaches zero as π‘₯ increases.

Part two says, similarly, what is the end behavior of the graph as π‘₯ 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 π‘₯ equals zero and move to the left. We’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 𝑦-value is increasing. It’s coming up. So, let’s think about our function 𝑦 equals one over π‘₯. If we plug in negative 10 for π‘₯, 𝑦 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, 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 π‘₯ and we’re plugging in negative values for π‘₯, 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 π‘₯-axis, and the π‘₯-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 π‘₯ 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 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 ∞. And the graph going down is then approaching negative ∞. And so, we can say the value of 𝑦 approaches negative ∞ when π‘₯ gets closer to zero from the negative direction, and that 𝑦 approaches positive ∞ when π‘₯ gets closer to zero from the positive direction.

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

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