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.

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with asymptotes. This problem says to sketch the graph of the following function and all of its asymptotes using a graphing utility. We're given the function F of X is equal to the quantity of 8 X2 + 2X minus 2 in quantity, divided by the quantity of 4 X minus 5. So we're going to begin by looking for the asymptotes of this function. So we're going to be looking for vertical, horizontal, and oblique asymptotes. So we're gonna start by trying to find any vertical asymptotes. So we call for a vertical asymptote to exist that our denominator needs to be equal to 0 at some point. So our denominator in this case is 4 X minus 5. And we need to set that equal to 0 and solve for X. So to solve for X, we'll need to add 5 to both sides. And then divide by 4. From on both sides and that will isolate X. On the left hand side of that equation, so we'll have X is equal to 5 divided by 4. And so this is the location of our vertical asymptote. So now we need to move on to finding any horizontal asymptotes, and we note that the degree of our polynomial in the numerator is actually greater than the degree of our polynomial in the denominator. That means we will not have any horizontal asymptotes. However, the degree of our polynomial in the numerator is actually just one greater than that in the denominator. So that means that we can have oblique or slanted asymptotes. And so we need to find the equation for that um oblique line. And to do that, we're going to do long division by dividing our denominator into our numerator. So, we're going to have 4 X minus 5. Dividing into the quantity of 8 X squared. Plus 2 X. -2. Here we just need to do long division. So, 4 X will go into 8 X squared. 2 X times, so we'll multiply 2 X by the quantity of 4 X minus 5, and that will give me 8 X squared. Minus 10 X, and I'll need to subtract this quantity from quantity of 8 X2 + 2X minus 2, and that will give me 12 X. -2. So now, we'll have 4 X going into 12 X 3 times. That means our quotient is going to be 2 X + 3. And so I'll multiply by 3 by the quantity of 4 X minus 5, and that will give me 12 X minus 15. And I'll subtracted that quantity from 12X minus 2, and this will give me 13 as my remainder, which I will ignore for an oblique asymptote. So, the equation for our oblique asptode is just going to be the quotient that I have. So that means the equation for our line for the oblique asymptote is going to be Y is equal to 2 X. Plus 3. So now I have all of my asymptotes identified. I can now use a graphing utility to graph our function as well as these asymptotes. And so when I do that, I get the following graph. Well, I have two dash lines indicating our two asymptotes. One is a vertical line at X equal to 5 divided by 4, and another one is a slanted line that follows the equation Y is equal to 2 x + 3. We also have plotted our function F of X, which is equal to the quantity of 8X2 + 2X minus 2 in quantity divided by the quantity of 4 X minus 5 on the graph as a solid line. So this is the graph that the question was asking for. 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.

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

Key Concepts

Asymptotes

Graphing Rational Functions

End Behavior

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