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.

$f (x) = \frac{4 x^{3} + 4 x^{2} + 7 x + 4}{x^{2} + 1}$

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with the asymptotes. This problem says to draw the graph of the following function and all of its asymptotes using a graphical utility, and we're given the function H of X is equal to the quantity of 2 XQ minus X2 + 4 in quantity, divided by the quantity of X2 + 2. So the first thing we need to do is identify all the asymptotes for this function, and we're going to be looking for vertical, horizontal, and oblique asymptotes. So we'll start off by looking for vertical asymptotes, and for a recall for the vertical asymptotes that we need the denominator to be equal to 0 at some point. But if I look at my denominator here, which is X2 + 2, X2 is always a positive number, and when I add 2 to it, that means I'm going to get a positive number that is always greater than 0. So therefore, our denominator can never equal 0, and that means that we have no vertical asymptotes. Now, if we look for horizontal asymptotes, we note that the degree of our polynomial in the numerator is actually greater than the degree of the polynomial in our denominator. So that means that we're going to have no horizontal asymptotes. However, the degree of our polynomial in the numerator is exactly one greater than the degree of our polynomial in the denominator. So we still could possibly have um oblique or slanted asymptotes. So define the equation for that oblique acitote, we're going to divide our denominator into our numerator using long division. So we're gonna have X2 + 2. divided into. The quantity of 2 x cubed minus X2. Plus 4. So to do long division, um, We're going to see how many times X2 will go into 2 X cubed, and that will go 2 X times. And so we will multiply 2 X by X2 + 2. And that will give us 2 X cubed Plus 4 X. And then we'll subtract this quantity from 2 X cubed minus X2 + 4. And when I do that subtraction, I'll get minus X2 minus 4 X. Plus 4. Then we need to see how many times X2 will go into minus X2. And that will go -1 times, so we will multiply -1 by the quantity of X2 + 2. And that will give me minus X2. -2, And then I will subtract that quantity from minus X2 minus 4 X + 4, and that will give me minus 4 X. Plus 6, and that's actually gonna be my remainder, which I can ignore for a uh oblique asymptote. And so the equation for the line that we're looking for is actually Y is equal to 2 X minus 1. So that is the equation for our oblique or slanted asymptote. And now we can actually use a graphing utility to draw our function, as well as the um the asymptote for this um equation for a line. So doing so, we'll get the following graph, where we have plotted. Our function H of X, which is equal to the quantity of 2 X cubed minus X2 + 4 and quantity divided by the quantity of X2 + 2, we applied that using a solid line on a graph, and it looks somewhat like a cubic function. And then we have a dashed line indicating the line that follows the equation Y is equal to 2 x minus 1, and that is our oblique asymptote. And so this is the graph that our question was wanting us to find. So I hope that this explanation has been useful, and I'll see you in the next video.

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.

$f (x) = \frac{4 x^{3} + 4 x^{2} + 7 x + 4}{x^{2} + 1}$

Key Concepts

Asymptotes

Graphing Rational Functions

Error Correction in Graphing

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