In Exercises 81–88, a. Find the slant asymptote of the graph of e...

Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+1)/x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the equation of the Slant as tote of the graph of the given function and graph, the given rational function following the seven steps, our given rational function is F of X equals in the numerator five X squared plus three. That's all divided by, in the denominator two X. Our answer choice A is the Slant Asom tote of Y equals five halves X. And then there is a graph answer choice B gives the Slant Asom tote of Y equals negative five halves X. And then there is a graph answer choice C gives the Slant Asom tode of Y equals negative five halfs X. And then there is a graph and answer choice D gives the Slant Asom tote of Y equals five halves X. And then there is a graph, we will go over these graphs in more detail. But first, let's work on the Slant Asom tote. So for the Slant Asymptote, we need to see if we have one by checking the degrees of our numerator and our denominator, the degree of our numerator is two. The degree of our denominator is one. And if the degree of the numerator is one more than the degree of the denominator, then we do have a Slant Asymptote. And in this case, yes, two is one more than one. So we do have a Slant Asymptote. Now to find out what the Slant Asom tote is, we're going to do the division here. We're going to take our enumerator five X squared and we're going to enumerate each degree of the numerator. So there are plus zero XS and then we have our plus three, that's all divided by two X. Now we do do the division. We ask ourselves what is five X squared divided by two X. And if we take a five X squared divided by two X one X in both the numerator and the denominator cancel out. And we're left with just five over two, excuse me, five halves X. And that is then multiplied back by our two X and five halves X multiplied by two X is the five X squared that we started with. Now we subtract, that's leaving us with no X squared and we bring down the zero X. So next, we ask ourselves how many times does two X go into zero XS? Well, it doesn't at all. So we put a plus zero, then we would multiply the zero by the two X and continue. However, we don't need all of that information for our slant Asymptote. We don't need the remainder we only need up until before the remainder, which is what we have here. The five halves X plus zero. And this is our slant Asymptote. It is the equation of Y equals what we have here. And we don't need the plus zero. So this is just five halves X, that's our slant as to and we got the first thing done. All right. Now, we just need to complete seven steps of our graphing. So let's move this on the page on the screen. I'm going to draw a rough sketch of a coordinate plane will have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. The first step in our graphing process is to determine whether there's any symmetry and we determine whether there's symmetry by putting in for X a negative X and seeing what we get. So we'll have a five multiplied by a negative X squared plus three in the numerator all divided by two multiplied by a negative X. This will simplify two. The negative X squared in the numerator is still going to be X squared. So we have five X squared plus three in the numerator all divided by a negative two X. And we could factor that negative out in front of the factor in the denominator. So this would be a negative five X squared plus three divided by two X. And that is the exact opposite of the original function that we started with. When we have the exact opposite, when it's the negative F of X, then that's symmetry about the origin. So we do have origin symmetry here for this problem. All right. Step one is done for step two, we are going to determine the Y intercepts. We determine the Y intercepts by finding F of zero, plugging zero in for X. So we'll have five multiplied by zero squared plus three in the numerator all divided by two, multiplied by zero. And the numerator simplifies to three. But the denominator simplifies to zero. And we can't have zero in our denominator that would be undefined. So there are no Y intercepts for this particular problem. Next step three, we're going to look for X intercepts. We find X intercepts by setting the numerator equal to zero. So we'll have five X squared plus three equals zero. We can subtract three from both sides to sulfur X. We have five, X squared equals negative three, divide both sides by five. We have X squared equals and negative 3/5 take the square right of both sides. On the left, we have X but on the right, we have plus or minus the square root of a negative 3/5. And those are all imaginary numbers. So because we're dealing with real numbers here with our graphing, we can't have imaginary numbers. And so there are no X intercepts either. No X intercepts, no Y intercepts for this one. So step four is going to be finding our vertical asymptotes. And we find those by setting our denominator equal to zero. So two X equals zero while solving for X here, that's just X equals zero. So we do have a vertical Asymptote at X equals zero. This is something we can draw on our coordinate plane. So X equals zero. That's the Y axis, we can draw a vertical, a Asymptote along the Y axis at X equals zero. All right, we got something. Next, we're going to look for any horizontal asymptotes step five. And we do that by comparing the degrees of our numerator and our denominator. And if that sounds familiar, yeah, we already did that. We found that the degree of the numerator was one more than the degree of the denominator. And when that happens, we have a slant asom tote. We don't have any horizontal asom totes, but we do have the slant asom tote and that's at Y equals five halves X, we can put that on our graph. So there, the, if this is in the format of Y equals M X plus B M is the slope B is the Y intercept, our Y intercept is at zero because there is no B so it intersects at the origin and then it's going to go for the slope five halves, it will go up five units to the right two units, up five units to the right two units. And from the origin, we can also go to down five units into the left two units continuing in the other direction. So we've got a relatively steep slope here and we can label this is our slant asom tote at Y equals five halves X. All right. So next step six is that we are going to plot points. And what we need to do is we need to look for points that are within the intervals between all of our X intercepts and our vertical asymptotes. And in this case, we only have one vertical Asymptote. So we have two intervals, one from negative infinity to zero and the other from pot from zero to positive infinity. So we only need to plot two X values. You can always do more, but this will tell us what's going on in those areas. So I'm going to choose negative one for the interval from negative infinity to zero and a positive one for the interval from zero to positive infinity. Plugging those in to our original ax rational equation will get a A Y value corresponding with the X value of negative one. The Y value will be a negative four. And for our X value of positive one, the Y value will be a positive four. Remembering that we had symmetry about the origin. So that makes sense. Now we can plot those points from the origin. We go to the left one unit for our X value negative one and down four units for our Y value of negative four approximately here. And then for our 0.14 from the origin, we go to the right one unit for X value of one and up four units for A Y value of four. So approximately here and step number seven is just too graph and our lines will be hugging up against our asom totes and within the regions where our plotted points are. So for the line that's going to be to the left of our vertical Asom tode of X equals zero, it's going to come from negative infinity for both the X and Y values to the right of our slant asom tote Y equals five halfs X. It's going to hug up against that and come up and hit our point of negative one negative four and somewhere around there, it's going to turn around and then come down along the Y axis and head toward negative infinity for the Y values. Now, our other line is to the right of our vertical asom tote of X equals zero. It's going to come from positive infinity for the Y value. It'll come down along our vertical asom tote of X equals zero. It'll hit our point at negative one negative four at some point around there. It'll turn around and then follow along our slant ascent tote of five halves, X heading toward positive infinity for both the X and the Y values. So there is our graph, we've got it. Now, we just need to compare this to our answer choices. So let's start looking at answer choices. A and B answer choice. A has a Slant Asymptote of Y equals five halfs X. That's great. But looking at the graph, they have drawn the lines in different places. So they have the first line to the left of the Slant Asymptote, Y equals five halves X and coming up along and it has an X intercept there and comes up along the Y axis. But our line is to the right of that Asom tote for the one that is to the left of our vertical ASOM tote. So we can rule out answer choice. A answer choice B has a slant as tote of Y equals negative five halfs X. So that one doesn't work either. Now, we can bring this work down to a different portion of our screen. And we will compare to these graphs for C and D. OK. So for answer choice C there's a slant, Asymptote of negative five halves X. So again, we don't have a negative slope for Asom tote. So that rules out answer choice C and answer choice D does have a slant asom tote of Y equals positive five halves X and the lines are drawn in the correct regions between the Asom totes. So for the, the line that's to the left of the vertical Asom tote at X equals zero that is below or to the right of the slant asom tote of five halves, X. And for the line that is to the right of our vertical Asymptote, X equals zero that is to the left or of our slant asom tote of Y equals five halves X. So this answer choice D does work well done. We'll catch you on the next one.

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+1)/x

Key Concepts

Slant Asymptote

Polynomial Long Division

Graphing Rational Functions

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