In Exercises 57–80, follow the seven steps to graph each rational...

Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=x^4/(x^2+2)

Accepted Answer

Hello, everybody have night today that we're going to look at this question that states graph the, the given rational function following the seven steps where the function is F of X is equal to three X to the fourth divided by open parentheses, X squared plus 13 closed parentheses. Now the estrogen provided all give us different graphs of F of X. Choice A has a parabola with a vertex at the origin and facing upwards. Answer choice B is a parabola with a vertex of the origin once again. But this time facing downwards answer choice C looks to be kind of two parabolas. They're not perfect Parabolas, but they have the same general shape. One has a vertex at the origin facing upwards and another one has a vertex below the X axis at a Y of around negative 15 and is facing downward. And answer choice D has again, what looks to be two Parabolas? They're not perfect Parabolas, but they have the gen the same general shape, one with a vertex at the origin facing downward and one with another one with a, with a vertex above the X axis at around a Y of facing upwards. Now for a question like this, where they ask us to solve for the graph with the seven steps, we have to keep in mind what those seven steps are and we'll simply just walk through them little by little step by step. Now, the first step, so step number one is to determine symmetry. So we will determine symmetry and to do that, we have to keep, keep a couple of rules in mind. So we have that if F of negative X is equal to F of X, then there will be Y axis symmetry. And I'll write sim just to abbreviate symmetry. Now, if we, another rule is if F of negative X and I'll rewrite my first F of negative X because the negative should go within the parentheses. So we'll have F of open parentheses negative inside of the parentheses and enclosed parentheses. So our second rule is if F of negative X is equal to negative F of X, then there will be origin symmetry. And once again, I'll write sim just to abbreviate symmetry. Now, for our case specifically, let's test this. So we have F of negative X will be equal to three, multiplying a negative X to the fourth power divided by a negative X squared plus three plus plus 13 in the denominator. And because each negative AX is being raised to an even exponent, so four and two, it will become positive. So we'll have three X to the fourth in the numerator and X squared plus 13 in the denominator. And this is the same as F of X. So as we said, if, if F of negative X is equal to F of X, there will be a Y X as symmetry. So therefore, we will have Y axis symmetry and that is our first result here. So now we go to step number two and this will be figuring out the Y intercept. And to do that, all we have to do is set our X equal to ze- zero. So we'll have F of zero is equal to three, multiplied by zero to the fourth power divided by zero squared plus 13 in the denominator. And we know that zero to any power will be zero and anything multiplied by 00. So we'll have zero in the numerator divided by 13 because zero squared is zero plus 13, 13 zero divided by anything is zero. So our Y intercept will be zero comma zero because we plugged in a zero for the X and we obtain a zero for the Y. So now we go to step number three, which is figuring out the X intercept. And in this case, we have to see what number we or, or we have to set our Y equal to zero. And because we are a fraction here, all we have to do is set the numerator equal to zero. Because if the numerator is equal to zero. Like we said, zero, divided by anything will be zero. And that will mean the Y is zero. So we just set three X to the fourth power equals zero, which means that X equals zero. Because if we divide zero by the three, we still have zero. And then X to the fourth power equals zero. So X must be equal to zero or the 0.0 comma zero which confirms our Y intercept of the origin. Now, finally, we move on to the fourth step here which will be figuring out the vertical, ask him to and I'm just going to draw a line between my steps. And I write that vertical ascent to can be abbreviated as va so for the vertical asma to all we have to do is set the denominator of our equation equal to zero and solve for that X. So we'll have that X squared plus 13 equals zero. And we do this because if our denominator is equal to zero, that means we will have an undefined value. Therefore, giving us a vertical as to now, if we have X squared equal to zero, X squared plus 13 equals zero, that means that X squared is equal to negative 13. And as we know, there is no number that once you square it will give you a negative answer. So no real number. And I'll scroll down a bit just so I have more space for my work. So no real number, therefore no vertical asymptotes. And that will be our answer for step number four. So we are now four steps in and we can move on to step number five, which is a horizontal acetose or H A as abbreviated. So for the horizontal ascent to we just compare the degrees of the numerator and the degree of the denominator. Now, in our case, the the green of numerator which is four. And to look at the degree, you just look at the power associated with the X in the function in the numerator, this is four. So we have three X to the fourth. So you look at the largest power and then in the denominator, our degree would be two because the largest power associated with the X is X squared. So we have the degree of the numerator is greater than the green of denominator, which is two. And this means that we will have no horizontal as and tilt. So when the de the degree of the numerator is greater than the degree of the denominator, there will be no horizontal asymptotes. Now we can move on to step number six, which is where we start plotting or plugging in certain points to see what other, what our graph would look like once we get there. And to do that, I like to do in X Y T. So I'll have a T shape where I can have the X on the left and the Y on the right. And for the X values, I'll plug in points. You can plug in any point that you would like or you can do the same thing for the Y. Let's say if you want a specific Y, what X would you have to plug in? I'm going to plug in specific points that I know will give me complete wise or whole number wise. So I have negative 2.908, negative 2.358 zero and then positive 2.358 and positive 2.908. So when I plug in negative 2.9 oh eight, as my X, my Y will be 10. When I plug in negative 2.358 for my X my Y will be five. If I plug in zero, we already obtained that the Y would be zero. That's our origin and our X and Y intercept. If I plug in a po positive 2.358 R Y will be five once again. And if I plug in 2.9 oh eight, our Y will be 10 once again, which makes sense because of what we said that we will have Y axis symmetry. So the Y values should be symmetrical when you go between negative X values and positive X values. So we've completed steps 1 to 6 and now we're just missing step number seven and I'll scroll down so I can complete it with more space step. Number seven is the final step here, which is simply to graph. So we'll draw a Y axis and an X axis and I plug in dashes on the X axis at X of one and except two and an X of three. And I'll do the same for the negative side. So I'll have a negative one, X, A negative two X and a negative three X. And for the Ys, I'll draw a dash at A Y of five and A Y of 10. So now I can plug in different points, I'll plug in a point at the origin which was zero, comma zero. Then I know that around two point between two and three and before half before 2.5. So at 2.358, we have a Y of five and then closer to an X of three will have a Y of 10. So I'll put a point there and those points are reflected on the around the Y axis. So we'll have a point at negative 2.358 and negative two point nine oh eight. And then I can just connect those points and we draw a curve and we see that we obtain a para Parabola as our answer. Of course, it doesn't look like a perfect problema because I'm doing it handwritten. But everything otherwise tells us that there is a problem. We have a Y symmetry, y axis symmetry, we start out of vertex uh at the origin and everything is reflected about the y axis. So with this in mind, we look at our answer choices and we see that answer choice A is a para at the vertex. And this is the only answer choice that has this. So that means that answer choice A is A is our graph and will be the answer to this question. So I hope that was helpful. And until next time.

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=x^4/(x^2+2)

Key Concepts

Rational Functions

Graphing Steps

Asymptotes

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