how to find end behavior of a rational function This is a topic that many people are looking for. bluevelvetrestaurant.com is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, bluevelvetrestaurant.com would like to introduce to you Easy Method to Find End Behaviour of Rational Functions. Following along are instructions in the video below:
Kumar and in this video. Ill show you a very easy method of finding finding and behavior from the given equation. So we need to find and behavior from equation.
We have two different equations here i have taken rational functions in this particular video. The first one is a rational function. 5.
Over x plus. 1. The other one is a function 1 over x square plus.
1. You need to write down the end behavior of these functions from the given equation. Now what is end behavior end behavior is the behavior of the graph of the function as x becomes negatively large value or x becomes negatively or positively.
Large value so we could write it like this x approaches negative infinity. That means. It is approaching a very large negative value.
What happens to y what y approaches that is what were trying to figure out. And what happens. When x approaches positive infinity in that case.
What value does by approach right so. This is what were trying to figure out for the given equations now one method of course is to graph so what you can do is you can always graph so. When you sketch a graph you can see what value it approaches let me show.
You a graph of this function. It is vertical asymptote.
At minus 1. Right. So.
That is how it is innominate. If i write x equals 2 minus 1. It becomes.
0. So that gives you a vertical asymptote so this vertical asymptote is for x equals. To minus 1.
Now. We know. It is a translated reciprocal function.
So if i write x equals to 0. What do i get for 0. We get 5 over 1 which is positive.
5. So that means this half on the right side is the positive half of this reciprocal function. So i could sketch a function like this right so that is the nature.
We know the characteristics of resolution. So we can approximately sketch the graph so from this graph. Which has a y intercept at five we can clearly see that as x approaches negative infinity.
Y approaches zero and as x approaches positive infinity y. Also approaches zero youll also notice that in the first case.
Y approaches. 0. From the negative side.
So at times. We write negative here and in the second case it approaches zero from the positive side so we write positive here right so that is not a must. But always a good practice to write like this so sketching a graph shows you the result and then easily you can write down the answer.
But at times. It may be difficult to sketch the graph. So what is the alternate alternate is calculate the value so what now we are going to do is we know negative large value could be like negative thousand right so if i replace x with negative thousand that is to see if i write what is the value of the function r of minus thousand right so r of minus thousand.
We can calculate so that will be 5 divided by within brackets we can write 1 minus thousand right bracket close is equal to we get a negative value convert that to decimals negative it is it is equals to negative 0005 so we can say it is a very very small value but it is negative in nature so that gives us the result 0. But from the negative sizes round to 0. Is that okay similarly you could also calculate what is the value of the function.
4. Plus. Thousand so if i write plus thousand here then i get 5 divided by 1001 right so that is 1000.
Plus. 1. And that is a positive large value im a small value.
Which is zero point zero zero four nine something right very close to zero. But positive in nature. So that is how you would actually get the end behavior by simple calculations correct.
So well adopt this method to answer the second question. So lets calculate what is f of minus thousand.
And what is f of plus thousand right for the given function so in this case f of minus thousand will be you know thousand squared is going to be positive right so it is going to be 1 divided by within bracket. So we can write again a bracket minus thousand bracket close square so and then plus one bracket close so this bracket is for the outside term and that gives you a number which is very very small. But positive do you see that zero point zero zero zero something.
But it is positive right so you kind of get it it approaches zero. If i replace this word. Plus.
Thousand then what happens then also we get 1 divided by within brackets 1000. Square plus. One so well get the same value right.
Its kind of zero point zero zero zero. Something right were you very close to very very close to zero. So we could say that the end behavior here is that when x approaches negative infinity.
Y approaches zero. But from the positive side and when x approaches i mean okay let me write x approaches positive infinity. Y.
Approaches. Zero from the positive side. Some of you can sketch this graph also when you sketch the graph this graph is not very easy to sketch.
Okay now if x is zero you get one y. We get one now youll notice that always the function is positive right so. If you take some more values of x.
Which could be any real number youll get a graph. Which is kind of like this right so it is approaching zero from the positive side you see how much different these two graphs are so i want to highlight that if you slightly change the equation nature of the graph really changes right and second part of this is how easy it is to find end behavior. By evaluating value the function for a large negative number or for a large positive number especially for multiple choice questions it really helps i am.
Anil kumar. And i hope you love this day thank you and all the best .
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