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)=2x^2/(x^2+4)

Accepted Answer

Hello everybody I have act today today. We're going to be at this question that states graph the given of function following the seven steps where our function is F of X is equal to eight X squared in the numerator divided by open parentheses, X squared plus 49 closed parentheses in the denominator. Now the answer choices provided all show us different types of graphs of our apple X function with answer choice. A being kind of an upside down V shape or a curved V shape almost where both the left side and the right side extend infinitely to each side. And a horizontal Asymptote at Y equals negative eight as your choice being is the same as answer choice. A but flipped around the X axis. Answer choice C is a very different graph from the other answers where we have a uh we have a horizontal Asymptote at Y equals negative eight. And above that horizontal as to we have a parabola facing upwards originating from the origin and below, below the horizontal ax uh Amato, we have what appears to be kind of an upside down curved L shape and that will be on the right side and the left side as it breaks in the middle. Now, answer choice D is the same as a choice C but flipped around the X axis. So, or a question like this, they specifically ask us to use the seven steps of graphing. So what are those seven steps? Well, let's just go through them little by little and we'll get to our final answer. So our first step, our label as step number one, and I'll write it as determine symmetry and how do we do that? Well, there's a couple of rules that we should keep in mind when doing this. And I'll say those rules right now. So if F of negative X is equal to F of X, then we will have Y axis symmetry and our right sim as an abbreviation. And we also know that 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 as an abbreviation. Now, let's apply this to our case. So in our case, if we have F of negative X, that will be equal to eight. So what will the numerator will be eight, multiplied by open parentheses? Negative X closed parentheses, squared. And that will all be divided by open bracket and then open parentheses, negative X closed parentheses squared plus 49 close bracket. And now once we, if we simplify this a bit further, we have that the negative X is being raised to the second power, which is an even exponent both in the numerator and the denominator. So we'll have that F, we'll have that F of negative X, it's equal to eight X squared since the negative of the X went away divided by X squared plus 49 in the denominator, which is the same as our F of X function. And therefore, we will have Y axis symmetry. And that'll be our answer for part one. So that's our first step completed. And we can go ahead and move on to step number two, which is where we start talking about the intercepts in our graph. Specifically, step number two will deal with the Y intercept. So for the Y intercept, all we have to do is plug in zero for our X. So instead of having F of X, we'll have F of zero and that'll be equal to eight multiplied by zero squared in the numerator divided by zero squared plus 49 in the denominator which is equal to zero, divided by zero or zero, divided by which is equal to zero because zero, divided by anything is zero or the 0.0 comma zero, which is the origin and that'll be our Y intercept. And we can move on to step number three where we talk about the other intercept or the other possible intercept the X intercept. So for the X intercept, it's the same concept, except instead of plugging in 04 R X, we plug in zero for our Y. And to do that, since we have a fraction, all we really have to do is set the numerator equal to zero. Because if the numerator is equal to zero, once, once again, zero, divided by anything will be zero. So we just said our, our numerator equal to zero, which is eight X squared equals zero, which means that X squared equals zero, which means that X equals zero. And we have an X intercept at zero comma zero, which once again is the origin. So that is it for step three and we can move on to step number four. Now step number four is when we start talking about the possible asymptotes in our graph specifically, first, we will be talking about a vertical Asymptote or I abbreviate this as va now for the vertical as and so we have to set the denominator in our fraction equal to zero because if you are dividing by zero, that's uh undefined and that will lead to a vertical acidil there. So our denominator was X squared plus 49. So we said that equal to zero, which means that X squared equals negative 49. And this cannot be possible because if you're squaring a number, there is no way for you to get a negative answer unless you're dealing with imaginary numbers. And we're not dealing with that in this case. So no real answer. Now we have no real answer, we will therefore have no vertical ascent. So in this case, and that will be your answer for step number four. Now we can move on to step number five where we deal with the other possible as horizontal Asymptote. And once again, I'll abbreviate this, this time, I'll abbreviate it as H A now for the horizontal as so we have to compare the degrees of our fraction. And what is the degree? Well, the degree is the number associated with the X, the biggest exponent associated with the X, both, both in the numerator and the denominator. So first we look at the numerator. So the degree of the numerator is, well, let's look at our narrator. Our numerator was eight X squared. So we have our, our only X here is an X being raised to the second power. We have no other X. So our biggest power associated with an X in the numerator is two. So the degree of our numerator is two and the degree of our numerator of our, the degree of our denominator. Now we look at our denominator and that we will be, well, once again, we look at the denominator, we have X squared plus 49. So our only X term here is an X squared, meaning that the biggest power associated with an X in the denominator is a two. So the degree of a denominator is also two. So we have the degree of the numerator is equal to the degree of the denominator. And when this is the case, we will therefore have a horizontal Asymptote at, well, when the degrees are equal to each other, now we have to set, now we have to look at the coefficient in front of that X with that degree. So in this case, we look at the numerator, we had an eight X squared. So we have eight and we'll have the, that coefficient divided by the coefficient of the denominator that was in front of our X squared as well. In this case, the only coefficient is a one. So we have a horizontal at eight as at eight, divided by one, four, Y equals eight. So that'll be our answer for our horizontal ascent and we can move on to step number six, which is plotting points. Now, for this, I usually do a T chart where I have an X column on the left and A Y column on the right. And I'm just going to scroll down a bit. So I have more space to work with. So I'll write my T chart and I'll plug in points for X at negative 10, negative 505 and 10. And then I let me add an X of negative 10. I'll have a Y of 5.369. And I'm, I'm going to obtain all the Y values by simply plugging in each X term into our Apha X function. So if I plug in a negative five, I'll get a Y of 2.7 oh three. If I plug in an X of zero, we already know we're going to get a Y of zero, we obtain that through our intercepts. If I plug in an X at five, I have it Y at 2.7 oh three. And now I know because we said we will have Y axis symmetry. Once we pass that X of zero, all the values will repeat themselves. So an X of 10 will give me a Y of 5.369 again. And now we can move on to the seventh and final step. And once again, I'll continue to scroll down a bit just so I have more space. The seventh step is to graph. So I'll draw a Y axis and then X axis and I'm gonna point out the X s that I marked. So I'm gonna put a dash at what should be an X of five and an X of 10 and an X of negative five and an X of negative 10. Now, for my Y values, I'll write an A Y for five and A Y at eight. Since that's where we, we'll have a horizontal as we have A Y of eight. And I'll just put a dash at Y equals five to kind of guide ourselves of where points will be. Now we had points at X of negative 10 is 5.369. So a bit above five and X of negative five was 2.7 oh three. So kind of halfway between the, the zero or the origin where once again, we, we had a point. So it's kind of halfway between the zero and X of zero or A Y of zero and A Y of five. So I'll put a point there and then this will reflect. So I'm just going to plug in the same points on the right side. Now, at Y equals eight, I'll draw a horizontal dashed line to represent our horizontal Aceto. And I connect, I can connect all the points I have starting from the origin, I'll extend right and turn vertically upwards. And as I approach A Y of A, I'm going to have to extend to the right and not as much up but more so to the right because we can never pass that Y 08, Y equals eight and we can never touch it and that will reflect on the Y axis. So on the other side of the origin as well, I'll start turning up up up until we get close to A Y of equals negative eight. And then we turn to the left. So we, the final shape of our graph will be kind of a va curved V facing upwards above the X axis and with both sides to the right and the left extending infinitely to the right and the left with a horizontal aceto I A Y eight. So we just compare that to our answers. And we see that it's not D and it's not C, so I'll scroll up to see answers A and B and we see that the graph will match with answer choice B. So that's the answer that we will pick. They have a horizontal as to at Y equals eight. And it's a kind of a curved V which with both sides extending infinitely to the left and to the right. So I hope that was helpful. And until next time.

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