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−2)/(x^2−4)

Accepted Answer

Hello everybody. Everything. All right. Today, today we're going to look at this question that states graph the given rational function following the seven steps where the rational function is F of X is equal to open parentheses, X minus eight closed parentheses, divided by open parentheses, X squared minus 16, closed parentheses. Now, the antigens provided all consist of different graphs with a vertical Asymptote at an X of negative four and an X of positive four. So if we just take some time to look at answer choices, ABC and D, we'll see the different graphs that they provide us and I'll just scroll down on a bit. So we can see C N D as well. And now I'll scroll down even more simply so that I have enough space to work with for this question. So what are the seven steps? Well, let's go one by one and slowly we'll get to the answer. So the first step here is to determine symmetry and we have to ask ourselves, how do we do that? Well, we just have to recall the different rules that come with symmetry. So if F of negative X it's equal to F of X, then there will be Y axis symmetry. And I'll abbreviate symmetry as SIM and we also know that if F of negative X is equal to negative F of X, then that means there will be origin symmetry. And once again, I'll abbreviate symmetry as syn. Now, in our case, let's do the F of negative X and see what we get. So we'll have F of negative X is equal to. And just so we can recall, I'll write the F of X without anything first. So I'll have F of X is equal to X minus eight divided by X squared minus 16 in the denominator. Now, that means that F of negative X is equal to negative X minus eight divided by open parentheses, negative X closed parentheses squared minus in the denominator. Now, this will be equal to negative X minus eight in the numerator and the denominator will have X squared positive X squared minus 16. Now, neither of these functions are the F of X that we started with or negative Eva. Therefore, there will be neither Y axis or origin symmetry and that will be it first step number one. Now, for step number two, we look at the Y intercept. Now, for the Y intercept, we just set our X, we plug in zero for the X basically. So we'll have F of zero instead of F of X, we have F of zero. And that will be equal to, if we plug in zero for our X, we'll have zero minus eight in the numerator and the denominator will be zero squared minus 16. And this will be equal to negative eight divided by negative 16, which is equal to one half because the negatives cancel out and eight divided by 16 is one half, which means what Y intercept will be zero, comma one half. And that is your first step number two. Now we stay in the same concept of intercepts and we look at the X intercept. So for the X intercept, it's the same idea except now we set the Y equal to zero. And to do that, we know we have a fraction. So all we really have to do is send the numerator equal to zero because as long as the numerator is equal to zero, that means our final answer will be zero because zero divided by anything will be zero. So our numerator is um X minus eight. So we have X minus eight equals zero, which means that was eight, which means our X intercept is at eight comma zero. And that is it first step number three. Now, first step number four, we look at the vertical asymptotes or the va and the va will be the abbreviation that I'll be using. So for the vertical as to now we said the denominator equal to zero. And it's because we can't, we cannot have zero in the denominator because that means if we're dividing by zero, that will be undefined. Therefore, a vertical amato. So we set our denominator X squared minus 16 equals zero, which means that X squared equals 16, which means that X is equal to plus or minus four. So therefore, we will have a vertical atom tilt at X equal four and X equal negative four. So we have two vertical as toads in this scenario which match matches with the yes or choices. And that is it for step number four. So now we move on to step number five. So we looked at vertical ascent totes and now we look at horizontal atos or abbreviated H A horizontal Asymptote, abbreviated H A. Now to figure out the horizontal as we look at the degree of the numerator and the denominator in our fraction. Now the degree is just the largest exponent associated with the X. So in the numerator, we have an X raised to no power. So we have that the green of numerator, which is one in this case because we're not raising into any power. So we're raising into the power of one is less than. Now. We look at the and I say less than because the degree of the denominator is two because if we look at the X and the denominator, we have X squared, we have no other axes raised to any powers larger than two. So our degree will be two. So we have the degree of numerator one is less than the green oh denominator, which is two in this case. And when this happens, we have therefore a horizontal Asymptote at Y equals zero, which just means the X axis. So the X axis will function as a horizontal Asymptote. Now for step number six and the final step before we be, we begin graphing, we're going to plot some points. So we're go, I'm gonna have a T chart. This is how I like to write my points. So I write a T chart with X on the left column and Y on the right column and I'll plug in different axes. So I'll plug in an X of negative six X up negative five an X of 1.72, an X of zero, an X of five and an X of eight. So you can plug in any point that you want. As long as you know, it will give you a clear picture of what your graph should look like. Remember you'll have vertical Asma tos at X equals four and X equals negative four and a horizontal Asy tode at Y equals zero. So you need to really pick your points accordingly. Now, if we have an X of negative six, we'll have a Y of negative 0.7. If we have an X of negative five, we have a Y of negative 1.444. If we have an X of 1.2072, we have a Y of 0.467. If we have an X of zero, we already know we have a Y of 0.5, that's our Y intercept. If we have an X F five, we have a Y of negative 0.333. And we know that our X intercept. So if we have an X of eight, we have a Y of zero. So with all that information, we can now go ahead and do step number seven, which is graphing. So we've arrived in the final step. So we simply draw a Y axis or an A A Y axis and an X axis and we can start plugging in things that we know. So we know that there will be a vertical ascent. So at negative four and positive form. So I'm just gonna draw a dash line up and down those two values. So I draw a dash line at an X of negative four and an ax of positive forward. Now, for the Y axis, I will draw dashes at A Y of one, A Y of two and A Y of three, we know that our Y intercept is at 0.5. So we kind of measure between the X axis and what we label as number one, we kind of measure halfway. Now we know if we have an X of 1. R Y is actually less than 0.5. So it's 0.467. So our graph, after the, we have our Y intercept would go a bit downwards before going upwards again. Now remember there's a vertical as at four. So before we get to four, the graph curves upwards and gets close to four infinitely as it moves up but never touches it. And as we move to the left, same thing will move from the Y intercept will continue close to negative four where there is another vertical as to and we turn upwards infinitely never touching that negative form value. And that's what will happen between X negative four and X of four. Now we look at extending our graph beyond those two points. So we know that our X intercept was it an X of eight? So there will be a point there. And we also look at the other side, we have at an X O negative five and negative six. So at an X of negative five and negative six, we want to remember that we mentioned that there will be a horizontal aceto at the Y axis. So we look at the points that we have, we have negative six. At negative six, we have a Y of negative 0.7 and at negative five, we have a Y of negative 1.444, which means as we move from negative six to negative five, our graph actually went down. And if we have a vert glass into a negative four, we know that our graph will be turning. So we try to simulate that and we turn downwards and approach negative four but never touch it. And as we move up, we'll approach the X axis forever but never touch it as well. And as we look on the right side, we know we have a system at eight. So we cross eight and continue. And as we move left from the X eight X equal eight, we know we have a glass and to at four and we know that at an X of five, we had a point at negative 0.333. So we know that our graph will be turning downwards and approaching that negative four infinitely but never touching it. So now we see the three parts of our graph, we have two vertical asym toads, a curve in the middle and two curves on each side. So now we just have to compare this to the graphs that they provided us and pick the best one which in this case will be answer choice. A which is the graph that matches our graph divided into three parts vertical as at X of theta four and X equals four. The centerpiece is a curve facing up and each the left and the right sides approach infinitely to the right and infinitely to the left as well as up and down. 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−2)/(x^2−4)

Key Concepts

Rational Functions

Asymptotes

Graphing Steps

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