Oblique Asymptotes

Graph the rational functio...

Question

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = (x² − 1) / x

Accepted Answer

Below 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. Graph the rational function and its asymptotes. Y is equal to x2 minus 4 divided by X. Awesome. So it appears for this particular problem we're ultimately trying to create a graph of this rational function, and we're also asked to show its asymptotes on the graph itself. So with that said, first off, in order to solve for this particular problem, we need to determine the domain and vertical asymptotes. So we need to note by looking at our expression itself for Y for our function of Y, we need to note that the function will be undefined when X is equal to 0. So let's make a note of this. So when X is equal to 0, it will be undefined because the denominator will be 0. So therefore, there will be a vertical asymptote at X equals 0. So let's put a box around that. So X equals 0 will be a vertical asymptote. So now we need to determine whether or not there will be any oblique or slant or diagonal asymptotes. So since the degree of the numerator is one more than the degree of the denominator, we can expect that this particular graph will have an Weak asymptote. So by dividing the numerator by the denominator, we will be able to find that X2 minus 4 divided by X will be equal to X minus 4 divided by X. So we need to note that as X approaches infinity, or when X approaches negative infinity, we need to note that the term -4 divided by X will 10 to 0 or will go to 0. Fantastic. So thus, the asymptote will be Y is equal to X. So let's make another box around that. So Y is equal to X will be an asymptote. So now we need to switch gears and we need to focus our attention on the behavior near X equals 0. So focusing our attention on the behavior where X is equal to 0. So with that in mind, We need to note that as X approaches, so once again, we need to note that as X approaches 0 from the right, that will mean that this term So like I'll do a line, so that will mean that the term negative 4. Divided by X, so -4 divided by X will tend to negative infinity. So that will mean that Y will tend to negative infinity, and as X approaches 0 from the left, that will mean that the term -4 divided by X will tend to positive infinity and Y will tend to positive infinity, or rather, Y will approach positive infinity. So now, Our next step is we need to evaluate the function at the X values near X equals 0. So let us note now, so when X is equal to 0 and we're trying to evaluate our Values of X for when X is equal to curly brackets of -211.2, once again in curly brackets. Fantastic. So let us note that when X is equal to -2, that will mean that Y is equal to -2 2 minus 4 divided by -2. So when we evaluate that, we will find that Y will be simply equal to 0, and using the same behavior at our different values of X, we will find that we will produce the following points. Or rather coordinate points, so parentheses -1, 3, or semi so we'll have the coordinate point 1.3 in parentheses we'll have 13 coordinate point and then we'll also have parentheses 2.0 coordinate point will be obtained. And we need to plot these points and connect them with a smooth curve. We also need to note that the curve should approach the slant asymptote as X approaches plus or minus infinity and should approach the vertical asymptote as X approaches 0. So with that in mind, now that we have all the required information that we need to plot or graph, you can use whatever graphing software is easiest for you to use, but when you plug in your function for Y into your graphing software, you should be able to produce the following graph. As you can see, we have our Vertical asymptote for Y equals X, and then we also have our oblique asymptote, which we have X equals 0, and then we have our oblique asymptote represented in green. So we have our vertical asyote, which is represented by a yellow dotted line, and then we have our oblique ascentote, which is represented by a diagonal dotted green line. And then we have our curve. Represented in red a bolded red line with our different coordinate points listed for parentheses 2.0, 1.3, and 13 in parentheses and 2.0. As you could see, and our curves get really, really close to our ascentoptees, but they do not cross or intersect them. So essentially we use our ascentoptees to To properly graph our curve, so they're like a guide to help us graph our curves. And that's 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.

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = (x² − 1) / x

Key Concepts

Rational Functions