Graphing functions Use the guidelines of this section to make ...

Question

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x²/(x - 2)

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. Graph the function F of X is equal to X2 divided by 2X minus 6. Awesome. So it appears for this particular problem, we're trying to graph this function of F of X, and that's what we're ultimately trying to solve for. So first off, in order to help us graph this function, let us first determine the domain of this function. So we need to know and recall, based on the information that's provided to us, the function is undefined when the denominator is zero, so we need to take 2 X. 6 is equal to 0, and we need to rearrange this equation to isolate and solve for X, and when we do, using a little bit of algebra, we will find that X is equal to 3. So thus the domain of F of X is negative infinity, 3 in parentheses. Union 3 infinity. In parentheses, where parentheses indicate that it's an open interval. Now we need to determine the intercepts and symmetry. So in order to determine the X intercepts, as we should recall, we need to set F of X equal to 0, and the numerator is X 2, so X 2 is equal to 0, so that will mean that X is equal to 0 when we isolate and solve for X. So therefore, the only X intercept is going to be at 0.0. Fantastic. Now moving our attention to solve for any Y intercepts. So we need to substitute in a value of X equals 0 into our equation. So F of X, but instead of it being F of X, we're going to plug in a 0 for X, so we're going to say F of 0. is equal to 02 divided by 2 multiplied by 0 minus 6. Fantastic. So when we evaluate this expression, we will find that it's going to be equal to 0. Fantastic. So therefore the Y intercept is also at 0.0. Now we need to solve for the symmetry, so we need to, as we should recall a note, we need to check if the function is even or odd, so we need to replace the value of X with negative X. So we need to take F of negative X is equal to negative X to the power of 2 divided by 2 multiplied by negative X minus 6. Which when we simplify, this will be equal to X2 divided by -2 X minus 6. So since F of negative X is not equal to, so let's make a note of this, since F of negative X is not equal to F of X, which would be even for symmetry, nor is it equal to negative F of X, that would mean for odd symmetry. The function has no symmetry. So no symmetry. I'll just put the word no for symmetry. So no, and I'll put in some quotation marks on that, so no symmetry. So this quotation marks are on the S denote symmetry, so there's no symmetry. In this particular, for this particular problem, there's no symmetry. So now switching gears a little bit here, now we need to solve for the asymptotes and the and the end behavior of this graph. So focusing on our vertical asymptotes first, the denominator 2X minus 6 equals 0, when X equals 3, so therefore there will be a vertical asymptote at. X equals 3, so that's going to be our vertical asymptote, which I'll denote as VA. So let me recap here. So X, so this denominator, which is 2 X minus 6 is equal to 0, when X is equal to 3. So therefore, as a result, there will be a vertical asymptote at X equals 3, fantastic. So now we need to note that as X approaches 3 from the left, that's what that negative sign in the power means. So as X approaches 3 from the left, that would mean that Y approaches negative infinity, and this is because X2 divided by 2 X minus 6 approaches 9 divided by 0 from the left. And as X approaches 3 from the right, that will mean that Y, in this case will be approaching positive infinity, and as Y approaches positive infinity because X to the power of 2 divided by 2 X minus 6 approaches 9 divided by 0 from the right. That's what that positive symbol and the power means. That means it's approaching from the right. So now we need to solve for any oblique or slant asymptotes, if any exist, and since, as we should be able to recall note that since the numerator's degree is 2, Is exactly 1 more than the denominator's degree 1, we need to perform synthetic or long division to solve for the oblique slash slant asymptote. So we can rewrite the division as x 2 divided by 2 multiplied by X minus 3 is equal to 12 multiplied by X to the power of 2 divided by X minus 3. So now let us use synthetic division to divide X2 by X minus 3. So we can say, 3139, so 10039. So that means, as we should recall a note that the quotient is X + 3 and the remainder is 9. So thus we will have that X 2 divided by 2 multiplied by X minus 3. It's going to be equal to 12 multiplied by X + 3 + 9 divided by X minus 3. Which will be equal to when we simplify 1/2 X plus 3 halves plus 9 divided by 2X minus 6. Fantastic. So with that said, our next step now is we need to determine, or rather, we need to note that this term 9 divided by 2 X minus 6 will 10 to 0 as X approaches plus or minus infinity. So therefore, without a doubt, the oblique asymptote. Or the slant asymptote will be Y equals 1/2 X plus 3 halves. So now we need to determine the end behavior. So we need you to note that as X approaches plus or minus infinity, that will mean that F of X is going to Be approximately 1/2 X + 3 divided by 2. Fantastic. So that means for large absolute value of X, the graph will get closer to the line Y equals 1/2 X plus 3 halves. So now, our next step is we're gonna need to perform the first derivative to help us determine the increasing slash decreasing intervals. So with that said, we now need to find the first derivative. Which our first derivative is going to be F of X using the quotient rule, so we need to recall and use the quotient rule. So when we do that, we will find that F of X is going to be equal to X2 divided by 2X minus 6, and finding the first derivative, you will find that F of X is going to be equal to 2 X multiplied by 2 x minus 6 minus X2. Multiplied by 2, all divided by 2 X minus 6 to the power 2, and this will be equal to 4 x 2 minus 12 x minus 2 X 2 divided by 2X minus 6 to the power of 2. Which is going to be equal to when we simplify it down to its most simple form, I'm gonna skip a few steps. It's gonna be equal to 2 X multiplied by X minus 6, all divided by 2 X minus 6 to the power of 2. Now we need to solve for our critical points. So in order to do that, we need to set the numerator equal to 0, and when we do that, we will find that 2 X multiplied by X minus 6 is equal to 0. So that will mean that X is equal to 0 or X is equal to 6, which are both within our domain. So now, our next step is we need to figure out the sine analysis of F of X. So that means we need to focus our attention on the denominator, which once again, as we should recall, the denominator is 2 X minus 6 to 2, will always be positive, except that X equals 3, where it will be undefined. So the sign of FX will depend on this denominator. That so it will depend on focusing on the main part of our denominator, which is 2 X multiplied by X minus 6. So we need to note that for the condition where X is going to be less than 0, that means if we choose X is equal to -1, and if we plug in a value of -1 in for X, we'll get that 2 multiplied by -1. Multiplied by -7, it's going to be equal to 14, which is greater than 0, and for 0 is less than X and X is less than 3, if we choose X is equal to 1, that would mean to be 2 multiplied by 1, multiplied by -5, which is going to be equal to -10, which is less than 0. Fantastic. And for the condition where 3 is less than X and X is less than 6, if we choose a value where X is equal to 4, so if we say that X is equal to 4, then 2 multiplied by 4 multiplied by 2 is going to be equal to -16, which is less than 0. And for the condition where X is greater than 6, if we say that X is equal to 10, we can say that 2 multiplied by 10. Multiplied by 4 is going to be equal to 80, and 80, of course, is most definitely greater than 0. So thus, we can go ahead and write very easily that F of X is increasing on Negative infinity 0 and 6 infinity in parentheses, that means it's increasing, which I'll have an arrow pointing upwards, and F of X is decreasing on parentheses 0.3 and 3.6 in parentheses. So I'll do an arrow pointing downwards to show that it's decreasing. So we have local extrema. So now focusing our attention on local extrema, we have at X equals 0, that means the changing is increasing from decreasing. So it's increasing to decreasing, which gives a local maximum, so that means it's F of 0 is equal to 0, and that X is equal to 6, that means it's changing from decreasing to increasing. Which will give us a local minima of F of 6 is equal to 36 divided by 2 multiplied by 6 minus 6, which is gonna be equal to 6. So now we need to perform a 2nd derivative in concavity test. So in order to find the second derivative, we need to start from F of X is equal to 2 X multiplied by X minus 6, all divided by 2X minus 6 to the power of 2. So after applying the The quotient rule, as we should recall, the second derivative, which is F of X, is going to be equal to 9 divided by X minus 3 to the power of 3. So now, solving for the concavity, we need to note that for when X is less than 3, that will mean that X minus 3 to 3 is less than 0. So that will mean that F of X is less than 0, which means that it's going to be concave down, which I'll call it CD concave down. So for when X is greater than 3, and X minus 3 to 3 is greater than 0. That will mean that F of X is going to be greater than 0, and that will mean that the function is gonna be concave up, which I'll call Cu. So now we need to focus our attention on finding the inflection points, and an inflection point, as we should recall, occurs when concavity changes. So the only candidate for this particular case is going to be at X equals 3. However, X equals 3 is not in the domain. It is a vertical asymptote, so there are no inflection points within our domain of F of X. So now we need to draw our vertical asymptote, which is going to be at X equals 3, and plot our local extrema 0.0 and 6.6. So let's make a note of this, 0.0 and 6.6. And we also need to plot our oblique asymptote, which once again is Y is equal to 1/2 X plus 3 halves. Fantastic. And we need to note that at X equals 0, Y will equal 3 halves, and at Y equals 0, that will mean that it's gonna be zeros equal to 1/2 X plus 3 halves, which will mean that this is equal to when we solve for just X all by itself, so we rearrange to isolate sulfur X that mean that X is equal to -3. So therefore, without a doubt, the two points on the oblique asymptote are going to be Parents 0.3 halves and 3.0 in parentheses, and we need to plot these points on our graph and connect them through a line and draw a curve based on the curve's character. that we determined together. So when we do that, using all the information that we collected together and going back to plot everything onto our graph, we will achieve the following graph. So I've gone ahead and used some graphing software to plot our information that we determined together. And as you can see, we have our oblique ascenttote, which is a slant ascenttote, which is a diagonal line represented by this red dotted line, and we have the points -3.0. And we have the points 0.3 halves and 6.6 and 0.0. These coordinate points are plotted with red dots, and then we have our vertical asymptote represented in purple, and once again our vertical asymptote is at X equals 3. And then we have our curves that are following in between our asymptotes, but noting that they get very, very close to our asymptotes, but they never touch or cross or intersect them, and these blue curves denote our concave up and concave down conditions, and they follow the boundaries of our asymptotes that we discussed. And that's it. That is our final answer for the graph. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x²/(x - 2)

Key Concepts

Function Behavior

Critical Points and Extrema

End Behavior

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