Graphing Rational Functions (Precalculus - College Algebra 44)

[@rosepierce9382]

I just realized Professor Leonard probably watches the whole video while editing it to point out any mistakes he might've made.That's a lot of work! He puts in a LOOOOT of effort into every single video. He stands and lectures for 2 hours straight. How many people can do that without collapsing? Honestly, teachers are so underappreciated and especially good ones like Professor Leonard deserve ALL the good things in life. I cannot be more grateful to him.

[@nihalnerlige8739]

I'm not gonna lie, Professor Leonard is the actual GOAT, I've been confused on end behavior for 3 days now. this video finally cleared it up. Thank you so much!

[@Redexn]

Wow. The longest video in the series yet!!! It'll take me 3 hours to go through because I stop and pause to solve the problems on my own but I know it'll be worth it. Thank you, professor leonard.

[@abbeylinsky3237]

i know im studying too much when i realize i actually felt genuine excitement for this video like it was the season finale of my favorite tv show 😵‍💫

[@funmaster5249]

You know, the whole world might be locked down, and life is dangerous and confusing. But on the bright side, prof Leonard now has more time to make videos. Bless

[@thedebis]

This felt like the college algebra Avengers end game moment seeing how everything in the last 43 videos just clicked together. Thank you so so much for making this playlist!!!!! Can't wait to continue onwards!!!

[@sillymatilda]

Stopped by to say thank you for your time and effort Professor Leonard. Your video lectures helped me to pass Calc 1,2,Vector and Differential Equations in university. I can only wish universities/school were filled with more teachers like you.

[@utuberaj60]

Suoerb explanation Prof Leonard. Never seen so good examples (quite challenging) and doing every bit of it SO WELL. OMG i never had a teacher like you before. Keep going

[@Harshita_INFP]

Blessed to have you sir... You're the best maths teacher I've ever come across.❤️

[@Arafat-my6fe]

Thanks a ton, Professor. I've learnt a lot from your videos and still learning. I am from Bangladesh and can't thank you enough for the sincere coupled with tremendous effort that you put in your work 🙏

[@surojpaul14]

But Prof. at 1:05:50 when I take the bigger one means 2x²-5x+2/x²-4 = 2 then I got x = 2,,there is no mathematical nonsensical statement produce,, I got the intersection point, but in the other hand the easy one produce a nonsensical statement,, that's contradicts and when I'm going to put x=2 in f(x), that's undefined, (2, undefined). Please help me

[@ThurmanMotorsports]

Great question, but yes, you're not going to get the same answer if its not a cross. In this case you're getting x=2 back showing where (if at all) you're going to have a H.A. cross. But because you know there's a hole at x=2, you know that there wont be a cross at that point. Using the simplified version of the fraction is just a different road to get to the same conclusion. Its just in this case you're getting that "non-sense answer" where two different numbers equal each other. Again great question, because yes, first time trying it you're expecting to get the same answer for both the original function and the simplified version. Two different "roads" that lead to two different "destinations" but the end and final conclusions are consistent.

[@JJ-uj1wi]

@@ThurmanMotorsports Thank you for answering this! I was wondering about the same question too

[@LordRamen12]

This is absolute perfect timing i got a a exam on this tonight thanks so much

[@sandlertossone1813]

how on earth can you give a thumbs down on this? if it is because it is too long, good luck in the future doing higher level math cause you will never get through calculus without totally understanding this stuff

[@ds5375]

@33:40 This became completely confusing to me. How did he identify there was a positive shaped quadratic asymptote? I didn't follow how he determined the (2,4) coordinates of it either. Further, do we treat parabolic shaped asymptotes as horizontal, or vertical asymptotes (important for crossing behaviour). Finally, I do not understand why the functional parabola is negative oriented, since the coefficient of the reduced rational leading terms are positive!

[@georgesadler7830]

Professor Leonard, thank you for a massive video/lecture on Graphing Rational Function. This material uses all the graphical tools that students learned from day one in basic mathematics. Graphing can also be checked by using online graphing tools and the Ti-83/84 calculators.

[@rosepierce9382]

There you are again! Honestly at this point, I am looking forward to finding your comment under every video as I go through the Precalculus playlist. How are you doing? I'd love an update!

[@malekstudies0944]

1:05:40 P.leonard said that it doesn’t matter whether or not you use the main or simplified version of the function when calculating intersection of the horizontal asymptote, but when i used the main function, i got x=2, meaning that there is an intersection. 2x^2-5x+2=2(x^2-4) 2x^2-5x+2=2x^2-8 -5x+2=-8 -5x=-10, so x=2….. what’s happening here?

[@zeeshan7372]

No bro, u have to factor the denominator cuz difference of squares have 2 solution so it is same thing as simplified version

[@scopeless22]

@@Ahmad.Khaled_101 Oh that makes so much sense. I made the same error but never considered that it "crosses" at a hole so there would be no point. Thanks this helped a lot.

[@Ahmad.Khaled_101]

@@scopeless22 God bless you❤️ bro

[@DiddleSlip]

@@zeeshan7372 Even if you factor out the difference of squares you would get 2x^2-8

[@MrBeast5200]

I got an A on my differential equations midterm thanks to you 😊

[@Aycore2011]

This was better than my favorite movie. As always, thank you sir for the wisdom.

[@ethan.lam3290]

Thank you so much Professor Leonard. This video saved my life

[@dannysegundo4018]

Absolute legend. Thank you Prof Leonard.

[@MrBluemanworld]

Let's face it: math is completely fun and stimulating.

[@whitebeardpirates7551]

The Best For Ever, Thank you For This BIG Effort ♥♥

[@silverdove9911]

amazing video. great review for my college calc course this quarter.

[@touqeerafsar1549]

You are doing a great job. I don't know why I am paying at college... maybe just for a piece of paper.

[@rosepierce9382]

Congrats on 701K, Professor!🎈

[@hamdihamdi6256]

thank you really from bottom of my heart

[@Bishowthapa-uv5sv]

big salute for your efforts

[@kellystevens6464]

These lectures are so great!

[@scopeless22]

Can someone help at 1:06:20. If I do it with the reduced fraction, I come out with no cross, but if I do it with the original function I get a cross at x=2. I do (2x^2 - 5x +2) = 2(x^2 - 4), I then factor to (2x^2 - 5x +2) = 2x^2 - 8, the 2x^2 cancels out, so end up with -5x + 2 = -8, this can factor to 5x = 10, which reduces to x = 2. I don't know where i made a mistake as I should expect it to not factor down right?

[@HappyGardenOfLife]

You'd have to do partial fractions which you won't usually learn till calc 2.

[@bunny4173]

hi joesph ..thanks for asking i ran with the same problem .. the ans to this question is that when you do with original function you get intersection at x=2 and later on we find that we have a hole at x=2 the hole tells that there is nothing at x=2 so it becomes non sense statement about what we found earlier(intersection at x=2) .....if we do that in the simplified function where the hole is already cancelled we do find that it becomes a non sensical statement i hope it helps

[@michaelsimpson5240]

@@bunny4173 Ahh, thanks that make sense now.

[@chacklidell]

@@bunny4173this doesn’t really answer it, the hole is at (2,3/4) and when you use the original function it’s saying that the line crosses the horizontal asymptotes at (2,2). So I’m am still confused.

[@michaelmihov1651]

I'm stuck here, unfortunately, gotta watch it a few more times

[@myrnalopez4575]

At 34:48 it should be x^2 -4, not x^2 + 4

[@shaniyadav3823]

That's great job

[@franciseylim11]

This is much more important than Apple Keynote.

[@mwenzikataya1904]

thank youn for the amazing videos...May you please make videos on the laplace

[@NYCGemini2000]

When are you gonna get to trigonometry?

[@sharaudramey9336]

@1:00:22 😅

[@khushisinghal2316]

Thank you very much professor!!

[@rosepierce9382]

In the last problem, wouldn't the horizontal and vertical asymptotes be also negative? The even vertical asymptote goes down and the odd vertical asymptote is reflected about the origin. But I guess in the end it doesn't matter because the negatives would cancel out anyway.

[@shaniyadav3823]

Thank you so much I am from India

[@NovaWarrior77]

Preciate it sir!

[@thatomofolo452]

recall that for rational function R(x)=P(x)/Q(x)have no factor in common

[@yruheregoaway]

Why did none of my math profs look like this.. Id be a math genius by now 😏

[@mahmodmoghaddam.p9277]

very very good

[@agnessmulengah4095]

Thankyou aleast I got a B+ in calculus 🥺🥺

[@HandsomeRoom-yv3cb]

How to graph (x^5 + 5x +6)/(x+1), please.

[@dashaagafonova519]

купить или учесть?

[@pinkman17_]

24:15

[@srittampanigrahi4810]

How To find Polar Asymptotes Please

[@quant-prep2843]

take x intercept and divide it by standard deviation and volatility then normalise by mean and the median of asymptotes, then do monte carlo simulation on oblique asymptote, hope it helps

[@abdallahm2299]

58:18

[@dashaagafonova519]

они пользуются моим предметом Δx

[@dashaagafonova519]

я за луч!

[@dashaagafonova519]

(0; -65/7) мне отравляют ум 65*4/7

[@rameshsilva8212]

👍

[@芒果牛奶-r3d]

My guess is that professor is a Scorpio

[@hamzasehavdic]

why

[@AnilKumarAnilKumar-rh4lx]

Sir please video in hindi

[@shravans7184]

Are you out of your mind .... I know you are mocking ....... Hindi is not even an official language of UN..........you "Kupamanduka"

[@rosepierce9382]

How the hell would he, an American, that probably only speaks English, make videos in Hindi?