Graphing Rational Functions (Precalculus - College Algebra 44)

How to efficiently sketch rational functions by using Horizontal Asymptotes, Vertical Asymptotes, Oblique Asymptotes, End Behavior, X-Intercepts, Y-Intercept, and select points.

[rosepierce]

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.

[funmaster]

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

[utuberaj]

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.❤️

[surojpaul]

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

[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.

[abbeylinsky]

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 😵‍💫

[sandlertossone]

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

[MrBluemanworld]

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

[malekstudies]

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?

[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.

[michaelmihov]

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

[ds]

@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!

[yruheregoaway]

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

[MrBeast]

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

[touqeerafsar]

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

[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
Can't wait to continue onwards!!!

[NYCGemini]

When are you gonna get to trigonometry?

[LordRamen]

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

[Arafat-myfe]

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 🙏