Graph each rational function. See Examples 5–9.

Question

ƒ (x) = (16 x^{2} - 9) / (x^{2} - 9)

Accepted Answer

Welcome back. I am so glad you're here. We are asked for the following rational function to graph. Our given function is F of X equals in the numerator 25 X squared minus four. That's all divided by the denominator X squared minus 16. For our answer choices, we have four graphs, we will go into more detail comparing those graphs toward the end. But for now let's work on graphing our rational function. Move this to a different part of the screen where there's a little bit more space to work. And now the first step I'd like to take is to see if we can factor this. Sometimes we'll use this function exactly as it is and sometimes we'll use the factored version. So looking at our numerator, we recognize a difference of squares. If you need any refresher on how to factor a difference of squares, please check out previous lesson videos. I'm going to do this one quickly. So for our numerator that's going to factor as the quantity of five X plus two multiplied by the quantity of five X minus two. And in the denominator, that's also a difference of squares that factors to be the quantity of X plus four multiplied by the quantity of X minus four. And so we've got this factored. Now we're going to work on graphing, going to draw a rough sketch of a coordinate plane. We have a vertical Y axis, a horizontal X axis. And they come together at the origin in the middle step one from the textbook and from previous lessons on graphing, a rational function is to determine the vertical as some totes. And we do that by setting the denominator equal to zero. Well, here's where the factored version is very helpful. We set the two factors of X plus four and X minus four equal to zero, solve for X and we'll get one vertical asom tode at X equals negative four and another vertical asom tode at X equals a positive four. Now looking at our graph from the origin, we go to the left four units and draw a vertical asom tote label that X equals negative four. And from the origin, we go to the right four units and draw another vertical asom tote label this one X equals positive four. All right. Step one is done. Step two, graphing a rational function. We are going to look for horizontal or oblique asom totes. And we do that by looking at the degrees of our numerator and our denominator. Here's where the first function is more useful. We can see that we have a degree of two in the numerator and a degree of two in the denominator. And when the degrees of the numerator and the denominator are equal to each other, then we find the horizontal asom tote by taking the coefficient, the leading coefficient of the numerator divided by the leading coefficient of the denominator. So in this case, the leading coefficient of the numerator is 25 we're going to divide that by the leading coefficient of the denominator, there's an unwritten one. So our horizontal asom tote is at we write that as Y equals 25. So this is going to be different scales for our X and Y axes because 25 is a much bigger number than four, but I don't want to get too far off of the chart here. So from the origin, we'll go up 25 units and we will call right here at 25 units. We draw a horizontal asom tote and label that Y equals 25. Alright, step three, step three is we're looking for our Y intercepts, we find Y intercepts by finding F of zero. And so plugging in zero four X. So we can use the first one here. We have 25 multiplied by zero squared minus four in the numerator divided by zero squared minus 16 in the denominator. Well, anything with the zero there is zero. So we'll just have negative four in the numerator divided by negative 16 in the denominator that will simplify to a positive 1/4. So the Y intercept is where X equals zero and Y equals 1/4 we can label that that will be a very close to the origin from the origin. We go up one quarter of a unit. And again, I'm not drawing this exactly to scale. I'll label that at 0 1/4. OK. Step four, step four. For graphing the rational function is to find the X intercept or intercepts, we find the X intercepts by setting the numerator equal to zero. Here, we'll use the factored one. We've got the factor of five X plus two set equal to zero. And the other factor of five X minus two equal to zero. Well, we can subtract two from both sides of the 1st +15 X equals negative two divide by five. And we have X equals a negative 2/5. And for the other one, the only difference is we're adding two to both sides. So five X equals positive to and we get X equals a positive 2/5. So when Y equals zero, we have X intercepts at negative 2/ zero and positive 2/5 zero again, very close to the origin. So graphing these from the origin, we go to the left 2/5 of a unit and label that at negative 2/ 0. And from the origin going to the right 2/ of a unit and we will label that one at positive 2/5 0. OK. We've got our X intercepts. Now we'll work on step five and this is where we determine whether this graph will intersect the horizontal Asymptote because it's a horizontal Asymptote. In this case, we determine if it does intersect the horizontal Asymptote by setting our function equal to the Y intercept. In this case, the Y intercept is at 1/4. So we're going to take 25 X squared minus four all divided by X squared minus 16 equal to 1/4. Now, to solve here, we'll multiply both sides by the denominator. So we'll have on the left 25 X squared minus four equal to on the right 1/4 multiplied by the quantity of X squared minus 16, distributing that 1/4 on the right hand side, I'll bring down the left 25 X squared minus four equal to on the right 1/4 X squared and then 1/4 multiplied by negative 16 is going to be negative four. Now, if we add four to both sides, then those fours are gone and we have 25 X squared equals 1/4 X squared. Well, that's just simply not true. They're not equal to each other. So in this case, no, we do not intersect with the horizontal Asymptote. OK. Now, the last one step six step six is to plot points and we want to choose points that are on either side of every vertical asom tote and X intercept. So what I have chosen, you do not have to choose these same points to graph. But for the X values from negative infinity to our vertical asom tode of X equals negative four, I've chosen the point or the X value of negative five. And then from the vertical Asymptote, X equals negative four to our X intercept of negative 2/ 0. I've chosen the X value of negative three in between our two X intercepts. I've chosen the X value of zero, which is great because we already did that one. And then between 2/5 and four, I have chosen positive three. And then for the last value from X equals four to positive infinity, I've chosen the X value of five and so four X equals negative five. You plug that in for the value of X in our original function, you can definitely pause the video and figure that out. But I've already done this and that's gonna give us a Y value of 69. So from the origin, going to the left five units and then going up units. I mean that's just somewhere way up here. Definitely not to scale. All right now plugging in negative three for the X value, we're going to get a Y value of negative 221 7th, which is approximately negative 31. So from the origin going to the left three units and then going down 31 units, which is, you know, somewhere over here really, what's important at this point is knowing which quadrant they're in and what's going on there, we already have the point of zero. And that's the Y value of 1/4 that's the Y intercept. Now plugging in positive three for X, this is cool because there's actually symmetry here and that's going to give us a negative 221 7th for our Y value. So from the origin, going to the right three units and then down again, negative 221 7th is the Y value. And symmetry again, plugging in five, positive five for X value, we're gonna get 69 again for the Y value. And from the origin, we go to the right five units and then up 69 units. So we've got this where we can now plot it. So looking at our graph, we see to the left of the origin up in the second quadrant, we have a point at negative 5 69. And then what's gonna happen is our lines for the graph are going to be above our horizontal asom tote of Y equals 25. And it's going to head toward negative infinity for the X values. And it's going to come along this horizontal ASOM tode of Y equals 25. And eventually it's going to turn up and it'll come up along our vertical asom tote of X equals negative four. It will go through the point of negative 5 69 and head toward positive infinity for our Y values. Now to the right of the vertical asom tote X equals negative four. Our graph is hanging out down in negative infinity land for the Y values. It's going to hug up against our vertical asom tote X equals negative four. It will come through our point of negative three, negative 221 7th. It will come up and it will hit our X intercept of negative 2/5 0. It will turn around at our, well, it will hit our point of 0 1/4 but there's symmetry. So it does turn around there and then it will hit our other X intercept of 2/5 0 come down through our 00.3, negative 221 7th and head toward negative infinity for the Y values along the vertical asom tode of X equals four. Now to the right of our vertical Asymptote, X equals four, we'll have this coming down from positive infinity along the Y axis or the of, of the Y values not along the Y axis, but of the Y values along that X equals four asy tote. It will go through our point of 69. Then it'll turn and then it'll hug up along our horizontal Asom tode of Y equals 25 go toward positive infinity for our X values along that Asymptote. And there's our graph we did it. So now we just have to see which one this matches, grab as much of it as I can and probably shrink it a little bit. Move it to a different part of our screen. OK? Now comparing with these four graphs looking at, let's start with our asymptotes, we've got vertical asymptotes at X equals negative four and X equals positive four. Well, answer traces A and B have vertical asymptotes of X equals negative four and X equals positive four. But answer choices C and D have vertical asom totes of X equals negative 2/5 and X equals positive 2/5. Those are the X intercepts. So we can eliminate answer choices C and D based on that. Now, let's look at the differences between A and B. Well, we have a horizontal asym tode at Y equals 25. Answer choice A has a horizontal asom tode of Y equals negative 25. And answer choice B has a horizontal asom tode of Y equals 25. So answer choice B looks like it works. We can compare further intercepts and further um details of the graph that all looks good. It's got the correct X and Y intercepts in that zoomed in portion close to the origin and answer. Choice B looks great. Well done. We'll catch you on the next one.

Graph each rational function. See Examples 5–9.
$ƒ (x) = (16 x^{2} - 9) / (x^{2} - 9)$

Key Concepts

Rational Functions and Their Domains

Asymptotes of Rational Functions

Graphing Techniques for Rational Functions

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