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)=(3x^2+x−4)/(2x^2−5x)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the given rational function following the seven steps. Our given rational function is F of X equals in the numerator. We have seven X squared minus six X minus 16. That's all divided by our denominator, four X squared minus five X. And our answer choices are four graphs. We will explore the differences between those four graphs toward the end. But now let's get started on graphing are given rational function and moving to a part of the screen where there's a little more space. All right. So before we begin those seven steps, I want to quickly factor this out and see if we can simplify it. If you need any refreshers on factoring. Please go check out some previous lesson videos. I'm going to factor this very quickly. Our numerator will factor two, the quantity of seven X plus eight multiplied by the quantity of X minus two. And the denominator, we can only factor out an X. So it's X multiplied by the quantity of four X minus five. And there are no factors in common between our numerator and a denominator. But that's OK. We will use this factored version for some of our steps and use the original for other steps. So this was helpful. Now for step one of our seven steps to graph this is where we check for symmetry and we check for symmetry by plugging in F of negative X and seeing what we get. So let's plug in negative X in for X. We'll use the first fraction. So we'll have seven multiplied by not X but negative X squared minus six, multiplied by negative X minus 16, all divided by four, multiplied by not X squared but negative X squared minus five multiplied by negative X. Now we can simplify this. Well, in the numerator and the denominator where we have a negative X squared, that's going to be a positive X squared. So in the numerator, we'll have seven X squared, negative six multiplied by negative X is positive six X. And then bringing over our minus 16 all divided by now, we have four X squared and negative five, multiplied by negative X is plus five X. So let's see what we've got here. Our seven X squared and our negative 16 in the numerator retained their signs and our four X squared in the denominator retained its sign. But the six X in the numerator changed from a negative six X to a positive six X. And in the denominator, the negative five X turned to a positive five X. So what does this mean? Well, when we're talking about symmetry, if everything, if the whole thing changes its sign, we get a negative F of X that indicates symmetry about the origin. And if we get what we started with, if nothing has changed its sign, if we get F of X back then we have symmetry about the Y axis. But in this case, we have neither of those. So we have no symmetry here. Alright. Step two of our seven step process, we look for our Y intercepts, we do that by finding F of zero. So plugging zero and for all of our X values, so we'll use that first fraction. We have seven multiplied by zero squared minus six, multiplied by zero minus 16, all divided by four multiplied by zero squared minus five, multiplied by zero. We simplify this in the numerator will have negative 16. But in the denominator we'll have zero. And if you have zero in the denominator that's undefined, we can't have that. So that tells us there are no Y intercepts for this one step three, we're going to look for X intercepts. We do that by setting our numerator equal to zero. And here is where the factored version is very helpful. So setting our numerator of the quantity of seven X plus eight multiplied by the quantity of X minus two equal to zero, we can set each of those two factors equal to zero and solve. So seven X plus eight equals zero will subtract eight from both sides. So we'll have seven, X equals negative eight, divide both sides by seven. And we get one X intercept at negative 8/7. So we can write down that we have negative 8/7 0 as an X intercept. Now setting the other factor X minus two equal to zero, we'll add two to both sides and we have another X intercept at two. So our other one is at 20. All right. Step four is to find vertical asymptotes. And we do that by setting our denominator equal to zero. Again, we'll use the factored one. So X multiplied by the quantity of four, X minus five equals zero. Set, the two factors equal to 01 is X equals zero. The other four, X minus five equals zero. Add five to both sides. We have four, X equals five, divide both sides by four and we have X equals 5/4. Those are, are two vertical asymptotes at X equals zero N A X equals 5/4. Next step five, we'll look for our horizontal asom totes. In order to do that, we compare the degrees of our numerators and our denominator, numerator and denominator. And so in the numerator, we have a degree of two. In the denominator, we have a degree of two and they're the same when the degrees are the same, we are going to take the leading coefficients and divide them So the leading coefficient of the numerator divided by the leading coefficient of the denominator that's seven divided by four. We can keep that as 7/4. So we have a horizontal asom tode at Y equals 7/4. And now step six is to plot some points, we want to plot points choosing X values in between all of our vertical asy totes and our X intercepts. And so in order to visualize that better, I am going to work a little bit on step seven, which is to graph. So I'm going to draw a rough sketch of a coordinate plane with a vertical Y axis and a horizontal X axis. They come together at the origin in the middle first, plotting those X intercepts from the origin. We're going to the left 8/7 of a unit just a little bit over one and we'll plot our point at negative 8/7 0. For the other X intercept from the origin, we're going to the right two units. So we have that X intercept at 20. Now we can draw our vertical asom totes. We have one at X equals zero. That's the Y axis. So we can draw an ascent tote along the Y axis and label that X equals zero. And for the other vertical ascent tode at X equals 5/ that's a little bit over one. So from the origin, we go to the right, just over one unit and draw our vertical asom Tode and we can label that as X equals 5/4. Now, we can draw our horizontal asom tote even though we don't need that for plotting the points. But so is not to forget it. So from the origin, we need to go up 7/4 of a unit, that's almost two units. And we draw our horizontal has some tope, we can label that Y equals 7/4. OK. Now, so the intervals from which we need to choose our points are from negative infinity to negative 8/7 from negative 8/ to 0, from zero to a positive 5/ from positive 5/4 to positive two and from positive two to infinity, the points that I've chosen, you don't have to choose the same ones. But the ones I have chosen for the X values are negative two, negative one, positive, one, positive three halves and a positive three. Now we're going to plug all of those X values into our original function for X and you're welcome to pause the video, go ahead and do that. But I've already calculated these. So for an X value of negative two, that's a Y value of 12 13th. For an X value of negative one, we have a Y value of negative one third for an X value of one. That's A Y value of 15 for an X value of three halves. That is a Y value of negative 6th which is approximately negative 6.16. And for our X value of three, our Y value is 29 21st, which is a little bit over one. Now, let's plot those from the origin for the point negative 2, 12 13th, we're going to the left two units and up just under one unit for our Y value. And that's approximately here for the point negative one negative one third from the origin to the left one unit. And then down one third of a unit approximately here for the 0.1 15 from the origin to the right one unit. And then up 15 units for our Y value that's not to scale, but it's up there for a 0. halves negative 37 6th, we're going from the origin to the right three halves of a unit. And then down approximately 6.16 units approximately here. And from the origin for our 0. 29 20 firsts, we're going to the right three units and then going up 29 20 firsts of a unit, which is just over one. So approximately there. Now step seven is graphing, we are essentially just connecting the points that we just plotted. And these are typically going to um come up against our asom totes that we have. So from starting from negative infinity along our horizontal asom tote of Y equals 7/ we're heading toward negative infinity for the X values. And then that's coming along our asom tote of Y equals 7/4. And it's going to go through our first point of negative two, 12 13th, then come down through our X intercept of negative 870 and then through our next point of negative one, negative one third. And then heading down toward negative infinity for the Y value along our vertical ascent tote of X equals zero. There's our first line mines or than it truly would be. The next line is going to be between our vertical asymptotes of X equals zero and X equals 5/4. And this one's hanging out really high up in the positive values for our Y values. And so that's going to be all in the first quadrant. It's coming from a positive infinity for our Y value. It's gonna come down at some point. It's gonna touch our point that we plotted at 1 15 and at some point it turns around and heads back up toward positive infinity along our vertical asom tote of X equals 5/4. And that's positive infinity for the Y values. And there's our second line. Our final line is to the right of our vertical Asymptote of X equals five halves. It's coming from negative infinity for the Y value and coming up along this vertical Asymptote, X equals five halves. It'll go through our point of three halves, negative 37 6th, then keep coming up and go through our X intercept of 20. And then it will go through our point of 3 29 21st and then it will come along the horizontal asom tote Y equals 7/4 and head toward positive infinity for the X value. And there's our graph, we did it. Now, we just need to find the answer choice that matches that. So I'll bring this up and probably need to shrink it a little bit and bring this to a different part of our screen. OK? So now we can see our answer choices and let's compare them. So answer choice. A, let's look at the asymptotes. Let's first look for a horizontal Asymptote of Y equals 7/4. Well, answer choice A has a horizontal Asymptote of Y equals negative 7/4. So we can rule that one out. Answer. Choice B does have a horizontal Asymptote of Y equals 7/4. OK? We'll come back to that answer. Choice C does have a horizontal asom tode of Y equals 7/4. Great. We'll come back to that answer. Choice D that one has a horizontal asom tode of Y equals negative 7/4. We can rule out answer choice D OK. Now let's look at the vertical asymptotes. So they both have a vertical Asymptote along the Y axis at X equals zero. How about our vertical Asymptote of X equals 5/4? Well, answer choice B has a vertical Asymptote of X equals negative 5/4. So we can rule that one out. And comparing with answer choice C, it looks like the lines are all in the proper positions and with respect to the asymptotes and all three of the ascent totes are in the proper positions. So answer choice C works well done. We'll catch you on the next one.

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x^2+x−4)/(2x^2−5x)

Key Concepts

Rational Functions

Asymptotes

Graphing Techniques

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