Sketch a possible graph of a function f that satisfies all of ...

Question

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

$f (- 1) = - 2$ , $f of 1 equals 2$ , $f of 0 equals 0$ , $\lim_{x \to \infty} f (x) = 1$ , $\lim_{x \to - \infty} f (x) = - 1$

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. Which of the following graphs satisfies all the given conditions below? F of -2 is equal to 3, F2 is equal to -3, F of 0 is equal to 0. The limit as X approaches infinity of F of X is equal to -2, and the limit as X approaches negative infinity of F of X equals 2. Awesome. So it appears for this particular problem. So we don't have our other graphs showing right now, but essentially what we're trying to do is we're trying to create a graph that satisfies all the conditions provided here by the problem itself. So with that said, that's our final answer we're ultimately trying to solve or we're trying to create a graph that satisfies all of these given conditions. So with that said, Our first step, so first off, we need to plot the given points. So focusing first on F of -2, so F of -2 equals 3, this means that the graph will pass so our particular graph, it will pass through the point 2.3, noting that we're following the XY formatting for our coordinate plot. So we're trying, that's our coordinate. -2.3, so that's X. Y. So once again, F of -2 equals 3 means that the graph will pass through -2.3. And for F of 2 equals -3, that will mean that our graph will pass through 2-3. And finally, for our last coordinate point, where it's F of 0, so we're focusing on F of 0 equals 0. That will mean that our coordinate our coordinate will be drum roll at 0.0, so at the origin point. Awesome. So, with that said, we now need to plot the points onto our graphs. So I've gone ahead and used graphing software to get a digital representation. Of what our graphs should look like. So now we have our points at -2.3, and we have a point at 0.0, and we have a point at 2.3. But what do these two red dotted lines mean? So these two red dotted lines, our our horizontal asymptotes. So how do we figure out why we have these vertical. Asymptote sorry, these horizontal ascents. So as we should note for these horizontal asymptotes, which sometimes some graphs have vertical asymptotes, but for this particular prong, we do not have any vertical ascentopess, we only have horizontal. So as we should recall a note for the condition where the limit as X approaches cause of infinity of F of X equals -2, this particular Condition tells us that there's going to be a horizontal asymptote, and let's write this in red since our asymptotes are represented by red dotted lines. That means that we'll have a horizontal asymptote at Y equals -2. And similarly, since we have a for our limit as X approaches negative infinity of FFX equals 2, that means that there's also going to be a ver so it's going to be a horizontal asymptote at Y equals 2. So that's as we could see now when we plot these asymptotes onto our graph, which, as we should recall, All of our acetoes are always represented by a dotted line or a line with dashes in it. And as you can see, we have two horizontal asceoptees at positive, Y equals 2, and at negative, Y equals -2. So we have Y equals -2 down below, and then similarly up above Y equals 2. For our two horizontal asymptotes. So now, we, like I mentioned earlier, we need to note that there are no vertical asymptotes mentioned, since there's no conditions given to us that suggest the presence of a vertical asymptote, such as undefined points for F of X. So therefore, as a result, there will be no vertical asymptotes for this particular problem that we need to concern ourselves with. So now our next step is we need to focus on the continuity. So as we should recall a note, the function is likely to be continuous, meaning that there shouldn't be any breaks, jumps, or holes in the graph. So with that said, we can now officially sketch our graph. So let us start by connecting the points -2.3, 2.3, and 00 in a smooth in a smooth natural curve, and let us ensure that the curve transitions to approach Y equals -2. On the right, and Y equals 2 on the left, creating horizontal asymptotes. So with that said, when we plot this crap, So when we plot our graph, Looking at our official when we draw in our curve, we will see that we have from like as we were discussing from the right and left, as you could see, so for our asymptotes, it's very important that we draw our curve as close to the asymptotes as possible, but they don't touch. So that's how it, as we could see, so it like it curves and it straights up and it curves and it goes straight up towards our diagonal, because if we draw a straight line going through our coordinates at -2.3 and 00 and 2.3. And if we draw our curves connecting to our straight line, we will have our. Curves that approach Y equals -2 and Y equals 2. And that is it. That is how we graph this particular problem. So, our other graphs, so like when you're watching this video, you will probably have seen like a bunch of other graphs for multiple choice ones. So to know how you select so how you should know that you can select the proper ones. So the way to know that you've selected the proper graph is you need to compare. Your graph to the provided graphs, the sket like we need to compare sketches to confirm the following pieces of information. So we need to recall and note that the graph has to pass through -2.3 and 23 and 00, and the left end behavior will approach Y equals 2, and the right end behavior will be approaching Y equals -2. And as you should notice that some of the graphs do not pass through any of these points, and one of the graphs has holes, which holes are basically like, as you can see on our graph, we have red dots that are filled in. So if you see circles on your graph and they're like not. They're not like circles that are not filled in, so circles that are not filled in, those are holes, and because one of the graphs has holes in it, that cannot be the final answer. So the only answer that's correct will be this graph that we found here together, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.
$f (- 1) = - 2$ , $f of 1 equals 2$ , $f of 0 equals 0$ , $\lim_{x \to \infty} f (x) = 1$ , $\lim_{x \to - \infty} f (x) = - 1$

Key Concepts

Asymptotes

Limits

Function Values

Watch next