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)=2/(x^2+x−2)

Accepted Answer

Hello everybody. Everything right. Today, today we're going to be looking at this question that states graph the given rational function following the seven steps where the function is F of X is equal to five divided by open parentheses, X squared minus five, X plus six, closed parentheses. Now the answer choice provided are all different versions of this graph. With answer choice A being divided into three parts with a vertical ascent to at X equals negative three and X equals negative two as choice B is the same as A except we are flipping the graph about the X axis. Answer choice C is the same as a except we have moved to the positive side of the X axis. So we'll have a vertical ascent to at X equals two and X equals three. And answer choice D is the same as answer choice C except we are once again flipping across the X axis. Now, for this question, they specifically ask us to use these seven steps of solving a graph. So what are those steps? Well, we're going to work through them one by one and slowly but surely complete the question. So step number one is determined symmetry. So how do we do that? Well, to do that, we need to keep in mind two different rules. So we have that if F of negative X is equal to F of X, then there will be Y axis symmetry. And all right. S sim as the abbreviation for symmetry. 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, all right, Sim as an abbreviation for symmetry. Now let's apply this to our case. So in our case, if we have F of negative X, we have five divided by open bracket, open parentheses, negative X closed parentheses, squared minus five, open parentheses, negative X closed parentheses plus six closed bracket. Now, if we simplify that we have, we have that F of negative X is equal to five divided by X squared plus five X plus six, which is neither equal to F of X or equal to F of Nega or negative F of X. So therefore, we will have neither Y axis or origin symmetry. So when neither case applies, then that just means we won't have symmetry around the Y axis or the origin. And that is it for step number one. And we can go ahead and move on to step number two where we start talking about intercepts. And in this case, we'll specifically talk about the Y intercept. Now, for the Y intercept, all we have to do is set our X equal to zero. So instead of F of X, we'll have F of zero and that'll be equal to five divided by zero squared minus five, multiplied by zero plus six. All of that in the denominator. And that was simplified to five divided by six or the intercept zero comma five divided by six. And that is it first step number two N R Y intercept. So we can move on to step number three and talking about the other intercept, the X intercept. Now for the X intercept, it's the same concept as the Y intercept. Except instead of setting our X equal to zero, we'll set our Y equal to zero. Now, for our function where we have a fraction, all we would have to do is set the numerator equal to zero and solve for that because zero divided by anything will be zero. However, in this case, our numerator is a five with no X. So there is no way that we can obtain a Y of zero. There is nothing. If we plug in any value for our denominator, we won't get any Y of zero because five divided by nothing will give us zero. So we would have to have zero in the numerator. And we don't, therefore, there will be no X intercept since we will never touch zero on our X. And that is it for step number three. And we can go ahead and move on to step number four where we talk about the vertical ascents and I'll abbreviate this as va so for the vertical as and so all you have to do is set the denominator or fraction equal to zero. And this is because if you set the denominator equal to zero, then we know we can't divide anything by zero because that will be undefined. So that will lead to a vertical glass until, so we'll say our denominator equals to zero. So we'll have X squared minus five, X plus six equal to zero. So we just have to graph this. So I'll have an open parentheses, a space and a closed parentheses. And again, I'll have an open parentheses, a space and a closed parentheses equal to zero. So now I'll fill in the parentheses. Now, I know that both parentheses need to have an X because they need to be multiplying to get us the X squared. And now we just have to find two numbers that when multiplied by each other will give us plus six or positive six, which is the last term in our equation. But when added with each other, we'll be able us negative five, which is the middle term in our equation. And those two numbers will be negative two and negative three. So we'll have X minus two in one parentheses and X minus three in the other parentheses. So just to confirm that. What if we have negative two, multiplied by negative three, we have positive six, which is the last term. And if we have negative two minus three, we have negative five, which is the middle term. So now we just set each parentheses equal to zero because since they're multiplying as long as one of them equals zero, then the final answer will be zero. So we'll have that X minus two equals zero and X minus three equals zero. So have X equal two and X equals three. And those will be our vertical ascents. And that will be it for step number four. And we can move on to step number five. So now we talk about the other Asymptote left the horizontal Asymptote. And I'll abbreviate this as H A now for the horizontal as and so we just have to compare the degrees of deum and the denominator. And how do we do that? So what is the degree? Well, the degree is the exponent, the largest exponent associated with an X. So the degree of the numerator, we look at the numerator and we see that we have no X. So the degree of the numerator will be zero since there is no X and then we look at the degree of the denominator. So for the denominator, we look once again at a fraction and we see that we have an X squared and A five X. Those are our R X terms X squared has a large, larger exponent than five X five X isn't raised, that X is not raised any exponent. So it would be an exponent of one X squared is based an exponent of two. So our degree will be two, we choose the largest exponent. So we have that the degree of the numerator is less than the degree of the denominator. And therefore, when this is the case, we'll have a horizontal Asymptote at Y equals zero or the X axis. And that is it first step number five. And we can on to step number six, which is plotting points. Now, I'm going to scroll down a bit just so I can have more room to work with. Now, for this step, I like to do a T chart where I have X on the left column and Y on the right column and I'll plug in different X values. So first, I'll plug in an X of negative one and I'm plugging this into our F of X function. So instead of having F of X, we would have F of negative one and so on. So if I have an X of negative one, I'll have a Y of 0.417. If I have an X of zero, I'll have a Y of 0.833. If I have an X of one, I'll have a Y of 2.5. If I have an X of 2.5, I have an A Y of negative 20. If I have an X of five, I'll have a Y of 0.8 33. And if I have an X of six, I'll have a Y of 0.417. So once again, I'll scroll down a bit just so I can have more room and I'll keep the, that chart on the screen just so we know the points that we have because for our final step, we have to grab. So the seventh and final step is to grab. So for this, I'll draw a Y axis and, and X axis and then I can start plugging in points that I know. So I know I'll have a vertical ascent to at two and another one at three. And I also draw dashes at 45 and six. Since those are XS that I tested now for the negative side of the X, I'll have an X at negative one and I'll draw another one at two and another dash at negative three just so we can look at ourselves for the Y axis, I'll draw a Y at one to and three. And for the negative side, I'll draw one way low at around negative 20. So first, I'll draw the vertical ascent toads so that, that will be represented by dashed lines up and down our graph at two and at three. And then I can start plugging in some points So I know that for an X of zero, we had a Y of 0.833. So almost one for an excel of negative one, we had a while around 10.5. And then we know, so we know that graph from negative 1 to 0 is going upwards. And since we continue to go upwards as we approach the tool which is a vert glass and so we have to turn quickly upwards. So we never touch that two volume. And as we go to the left past ne the X of negative one to the left, we know that we will have a horizontal aceto at Y of zero or the X axis. So we'll turn quickly to the left and avoid the X axis forever. So that first shape on the left hand side of our graph will be almost a curved L L mirrored curved L. So now we look at in between the two and the three, the X two and X three. And that's where we plugged in an X of 2.5, which mean, which gave us a Y of negative 20. Now, we know we, we don't have a, an X intercept. So that means our graph cannot be going upwards. So from that negative 20 as we go to the right towards three, we curve downwards. And as we move to the left towards two, we also curve downwards so that we always avoid the X axis So the middle portion will look like an upside down U with a vertex at negative 20. And now we can look at uh the right side of our graph or the points past the vertical ascent. So at three at X equals three, and we can see that we had, at an X of five, we had a Y of 0.833 at an X of six, we had a Y of 0.417. So we know that from that five to the sixth, the graph is going down. And once again, we know that the horizontal ato is at Y equals zero. So we continue to go right, but very slowly going downwards avoiding the X axis. And as we go to the left to the left of the X equals five, we know that the graph is going upwards because from six, from X equal six to X equals five, the graph goes up. So we continue that trend towards our vertical as at equals three and we continue to turn up, up, up to avoid that X equals three. And that will be what our graph will look like. That side will look like a curved L. So now that we have this information, we can compare this to the choices provided. And if we just look right and on the screen, we'll see that a choice D has the graph that we just drew vertical ascent tos at X equals two and X equal three. The right side of the graph looking like a curved L and the left side to the left of the vertical at X equal two kind of looks like a mirrored image of that or very close to it. And between the vertical ASOM at X equals two and X equals three, we have an upside down U with which with a vert vertex at a Y of negative 20. 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)=2/(x^2+x−2)

Key Concepts

Rational Functions

Finding Asymptotes

Intercepts

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