Complete the following steps for the given functions.

Question

c. Graph $f$ and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f (x) = \frac{x^{2} - 3}{x + 6}$

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Sketch the graph of the following function and all of its asymptotes. GFX is equal to 3 multiplied by X2 + 2 all divided by 2 multiplied by X minus 4. Awesome. So it appears for this particular problem, we're asked to sketch a graph of this particular function, so we're gonna be creating a graph for this function of GFX. Awesome. So now that we know that we're trying to create a graph for this function of GFX, our first step that we need to take is we need to identify all of the vertical asymptotes firstly. That's our first job that we need to accomplish. So in order to identify the vertical asymptotes, we need to recall and note that vertical asymptotes occur when the denominator is equal to 0 and the numerator is not 0 at the same point. So with that said, we need to set the denominator equal to 0 and solve for X. So when we do that, we need to take 2 X or 2 multiplied by X minus 4 is equal to 0, so now we need to rearrange this equation to isolate and solve for X using a little bit of algebra. And when we do that, we will find that X is simply equal to 2. Awesome. And our next step now is we need to check if X equals 2 does not make the numerator equal to 0. So in order to do that, we need to take 3 multiplied by x 2, which will just say 3 multiplied by x 2. And you you could add a little multiplication symbol, like a little dot in between those two numbers, but you don't have to. When you take 3 multiplied by X2 + 2 is equal to 3, plugging in our value for X, which is 2 to the power of 2. Plus 2. Is all equal to. 3, and then we have to note that 2 to the power 2 is gonna be 2 multiplied by 2, which is 4. So this will be all equal to, when we plug this into our calculator, simply 14. So, therefore, without a doubt, because we have determined that when we plug in our value for X equals 2 into our expression that it will yield 14, so that will mean that there is a vertical asymptote at X equals 2. So that means we have once again a vertical asymptote at X equals 2. So let's put a box around that, use that as one of our asymptotes. So now, our next step is we need to identify any horizontal asymptotes if there are any. So, we need to recall and note that for rational functions, if the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote will be the ratio of the leading coefficients. So let's say that again, that's very important. So for any rational function, if the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is the ratio of the leading coefficients. So to be quick, like to quickly recap here, the numerator is our expression on the top part of our division bar, and the denominator is the bottom part of our division bar, just in case we do not remember. So with that said, as we should recall and note once again, the degrees of the numerator and the nominator are 2 and 1 respectfully, so there is no horizontal asymptote. Instead, we need to look for an oblique asymptote. What does oblique mean? Oblique is just a fancy word to say, a fancy way of saying diagonal. So we're trying to find a diagonal asymptote. So, and you'll probably hear from your teacher, your professor. Or your textbook will use oblique, but whenever they say oblique, they mean diagonal. So we're trying to figure out a diagonal asymptote. So, in order to identify this oblique asymptote or asymptotes, we need to figure out the degree of the numerator. So we need to recall a note that when the degree of the numerator is exactly 1 more than the degree of the denominator, and then we need to divide the numerator by the denominator. So in order to do that, we need to recall and use long division or synthetic division with our expression, which is 3 multiplied by X2 + 2 divided by 2 multiplied by X minus 4. So, doing our synthetic division, Outside of our division bar, we need to take 2 multiplied by X, so 2 X minus 4. So divided by, so we need to do our division box here. So divided by 3, so inside of our division boxes 3 X2 + 2. So, Ultimately, when you plug in our things, so the devices when you perform long division or synthetic division, So we will get, ultimately for our final answer, 3 halves X plus 3. So that will mean that the division gives us G of X is equal to 3 divided by 2 or 3 halves X. Plus 3 + 14 divided by 2 X minus 4. So that will mean, without a doubt, that our oblique asymptote will be Y equals 3 halves X plus 3. Awesome. So now our next step is we need to plot the asymptotes. So we need to draw the vertical asymptote as a dashed line, or rather a dotted line, because typically whenever you graph any sort of asymptotes, it's a dotted line, so you know it's an asymptote because it's indicated because it's dotted. So if you want to indicate an asymptote, you use a dotted line. That's the universal language for an asymptote. So, we need to draw our vertical asymptote as a as a dotted line at X equals 2, and our oblique or our diagonal asymptote. At Y equals 3 have X plus 3. So let's put a box around that because that's another one of our answers that we need to use. So, with that said, I've gone ahead and used graphing software to create a digitized and very clear diagram of our two different asymptotes here. So as you can see, we have our oblique asymptote, which is diagonal, and as we could see, The intercepts of these so the intercepts of this oblique asymptote, which when it says intercepts, that means it hits our Y axis and our X axis at two different points. So, So two different points it will strike. So our first point is on the Y axis at 0.3, where it's X. Y. So it's 0.3 on our Y axis, and it intercepts our X-axis at -2.0, as indicated in our figure here that we're using. Awesome. And as you can see, we just plotted our oblique asymptote, which is set up as a slope formula. So it's just a simple simple line. And as you can see, we have our oblique acetote which is in green, a green dotted line, and then we have our vertical ascetope, which vertical means up and down, and it lines up with the Y axis and it goes through X equals 2. Awesome. So, moving right along here. Our next step now is we need to determine the behavior near the asymptote. So, as X approaches to from the left, and let's make a note of this, so this is as X approaches 2 from the left. That's what that negative symbol and the power means. So as X approaches 2 from the left, the denominator will approach 0 from the left and will be negative. And since the numerator is always positive, the function GFX will approach negative infinity. So that will mean that when we take the limit, and let's make a note of this, the limit as X approaches 2 from the left of 3 X2 + 2 all divided by 2 X minus 4. This will mean it will be equal to 3 when we plug in our different, so when we plug in our value for X for the value of X as 2, so 3. Multiplied by 2 to the power of 2 plus 2 divided by 2 multiplied by 2 for approaching from the left. So 2 multiplied by 2 approaching from the left, minus 4, which will be all equal to 14 divided by 0 approaching from the left, which will be equal to negative infinity. And similarly, as X. So now we need to consider as X approaches 2 from the right. That's what the positive, the positive sign in our power means. So as X approaches 2 from the right, the denominator will approach 0 from the right and will be positive. And since the numerator is always positive, the function of G of X will approach infinity. So similarly, using the same exact equation as we did for our limit, but instead. We're gonna be plugging in to approaching from the right, and when we do that, that will mean that this will mean that it is approaching. Or it's going to be equal to infinity, positive infinity to be precise. So, as a result, the function will follow the line of the oblique asymptote at the far left. And far right of the graph. So now, with this information, we can officially sketch the graph. So using the asymptotes as our guides and additional points, we can sketch are two separate curves because as you can see, as we should be able to recall a note, we have a hyperbolic based like expression for our function, so we need to plot two separate curves inside of our two asymptotes. So really quick here, I'm gonna write down our limit, just in case we don't get confused, we need to take the limit. As X approaches 2 from the right of 3 X2 + 2 divided by 2 X minus 4. And this will be equal to 3 multiplied by 2, so 3 multiplied by 2 to the power 2 plus 2 divided by 2. Multiplied by 2 approaching from the right minus 4, which will be equal to 14 divided by 0 approaching from the right, which is equal to positive infinity. Because this is important because you need to be able to write your two separate limits down for your professor, your teacher, so they could see that you know the process to plot these correctly because they want you to show all of your work, to get all the points possible to make sure you get the correct answer. So, with that said, Using our information, when we plot our two curves, we will achieve the following graph. And as you can see, our two curves are plotted in red, and it's very, very important that when we plot our curves, they will get very, very close to these asymptotes, but they will never touch or cross them. So they're going to be basically getting closer and closer and closer to the asymptotes, but they will never touch them. So, and once again it's important that we use our asymptotes as guides and the additional points we need to sketch our curve following these asymptotes, and once again they'll never touch or cross them. So make sure when you sketch these that you don't cross over to the other side of the acetote or have it looking like it's touching the asymptotes, or else you might lose points, grading wise. Awesome, and that is it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Complete the following steps for the given functions.

c. Graph $f$ and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f (x) = \frac{x^{2} - 3}{x + 6}$

Key Concepts

Asymptotes

Graphing Rational Functions

End Behavior of Functions

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