Introduction to Graph Transformations (Precalculus - College Algebra 14)

[@Tirthankar17]

We should all be thankful that people like him exist in this world.

[@ammishtandon4961]

I had my Calculus 3 exam this week and I rocked it because of your lectures..You are phenomenal.. You teach with so much interest and passion ,and not like any dull teacher, which motivates us as well to study with passion and to be honest your lectures made me fall in love with calculus...I'll be taking differential equations in the next semester, so I request you to please continue that as well when you have time..Hats off to an amazing teacher..I TRUST THE LEONARD!

[@TwDjLsT]

GET SOME Mr. Tandon.

[@johnbiluke8406]

Absolutely, comical.

[@averageyoutuber-kk5ly]

out of all my math teachers from various colleges and high schools, you teach math how it is suppose to be taught. not just how to do things but why you do it and what is actually is. well rounded lessons = actual learning and not just memorization. 11/10

[@MahdiSadeghi-ly2th]

Dude I dont know how to thank you. You are one of the very few people who really care about why something happens and dont wanna just memorize. Your lecs are an inspiration for other people to understand why logic and reasoning matters.

[@TheDemarie76]

THANK YOU! THANK YOU! THANK YOU!! I am a nontraditional student and I have never seen math like this. I have been staring at my 113 homework for three days trying to figure this out. This one video has helped me understand. I will be using any and all of your videos to finish my course. Thank you again. I have tears in my eyes from the relief of understanding it finally.

[@angelinastudy]

I am in high school in Russia. you explained the topic in 40 minutes better than my teacher in a week. Thank you

[@1Vice876]

I rememer watching Prof. Leonard videos two years ago when I was talking Calculus. Thank you🇯🇲🇯🇲🇯🇲

[@adomseddies1093]

I think I have found the mathematics teacher I had been looking for all this while. With much appreciations prof

[@kjpillay]

Best Math teacher on KZbin. Really grateful for the time you took to make these videos. You are helping generations of aspiring students. Not sure if teachers are aware just how much they change the world simply by doing what they do. Thank you Sir.

[@dr3dr3-nb9mp]

All I can say is thank you for teaching with passion and explaining why things are what they are. Lot of academia live and thrive on theory but forget before theory comes hypothesis and if there is hypothesis there is always an idea and this idea is needed by someone that was in your head when you formulate or observe it so you shouldn't expect him/her to just understand your theory which is what was used to verity and proof your idea. Your way of teaching has make me to understand, love and find a way to apply mathematics to many problem Ex is realizing this is how vector scalar, vector addition, subtraction, rotation, transformation is derived and use in graphics. This is a light bulb moment for me and now I just understand and conceptualize what I already have an idea of but now my idea just turned to deeper understanding this is huge for me. Phasing shifting, phase lock loop all used in digital electronics are derived from this concept and you just demystify it all in first 10 minutes of a video that's not even calculus or solving any equation yet but ideas of what to come. Thank you, thank you and thank you!!

[@kabulkhan2245]

I can’t even begin to explain how much your lectures meant to me..💓

[@reginavanata8958]

"You need to get your shift straight" haha.

[@NYCGemini2000]

hahah i laughed when he said that

[@bear532]

Literally paused to comment this. You beat me to it.

[@nm1978]

be nice now, listen and learn.

[@ConservativeHippie001]

I caught that too. Lol! So funny

[@tommywilken197]

@@bear532me too ^_^

[@skippytheelder]

It's amazing how you turn something that I always saw as complicated and you explain it in a simple yet completely accurate way.

[@CT-rw1rw]

came to your most recent video to comment this. currently taking calc 2 and have been watching your calc 2 playlist. Because of you, for the first time ever, i really enjoy doing math. We all appreciate you more than you will ever know. thank you

[@TabethaMartinHoward]

great professor me and my peers use his videos since our teacher is out because of injuries. he makes it really easy to follow and understand. i got a 87 on my exam just by following his professor! Wish me luck on the next exam.

[@shanesmith4614]

Prof Leonard, I'm really glad you are young because it would be great if you could do videos for all math subjects over time. For example, I made it through abstract algebra but didn't really learn much because they didn't explain it well. It seems really interesting the "if this then that" and the logic study behind proving things such as the square root of 2, etc. We really need your intelligence and excellent teaching even more for the abstract stuff. It would be great to have you do an abstract algebra video and continue on with higher level classes as you progress. Of course, I'm enjoying your videos as you progress through all math subjects starting with the fundamental classes and on upwards. Thanks and I hope you'll forever continue with all math classes and subjects up to upper division and advanced classes. Thank you for all!

[@haven216]

Holy crap, using the birth year analogy for the horizontal shift is GENIUS. Tysm for these videos!

[@AB-cn5hb]

Professor this transformation oddly enough helped me to understand linear algebra transformation. I really appreciate it.

[@AlytheAlleyWay]

This channel is so refreshing after STRUGGLING to learn and understand in my AP precalc class with my actual teacher. You truly have a gift for teaching, and I really appreciate the fact that you actually find new ways to think about concepts to make it easier, and not just go by the book. Thank you so much! I truly cannot believe I've found such an amazing teacher.

[@sfrank853]

The moment professor leonard started talking about the input value for function or output value for y which is outside of function - it literally makes everything clear to me for outside of function = ( 1,1) become ( 1, 1+k) for inside the function = the same (1,1) becomes 1+k for every value of x which actually makes the entire x axis translate +k unit. His explanation for key points for assessing the basic characteristic of graph is millions times better than all of the jargon i have gone through which involves mere emphasis of cramming on this notion boundlessly grateful for him for doling away such insightful, simple and intuitive lessons - infinite times better than any of the content i have gone through in class or on internet. effect of "Input value or output value" of the function is the main gist of understanding this

[@sharaudramey9336]

i think one of the simplest but overlooked concepts when looking at the functional expression is how important it was for me to understand that no matter which variable is reflected y, h, k, a or b as has been used over many of these videos is that everything outside of ( ) refers to output. Really internalizing that in how I look at this how made this seem very intuitive.

[@meyerschristina2]

For those asking about the last problem with -x inside the square root, after discussing with a classmate, I believe this is possible because for a negative inside the square root with a POSITIVE INPUT -(x) the outcome is imaginary. But for a negative inside the square root with a NEGATIVE INPUT -(-x) your output (y value) is positive and this half can be graphed about the y-axis. So: f(4) would be imaginary but f(-4) would give you (-4,2). I hope this helps :)

[@bunny4173]

yes you are exactly right

[@nathanmohr4259]

Thanks for leaving this explanation! I was confused for a second.

[@prashantsingh3664]

ohh man thnx , just a basic concept but was in my mind for a while . Lol

[@SeattleMommy8338]

Thank you so much!!! I haven’t studied this material in twenty years…now my seventh grader is coming to me wanting help with his homework!! 🥺😭

[@chad2476]

I had zero interest in math before, but thank you professor Leonard you made me love math!!

[@777ISR]

I really appreciate the way that you teach your subject matter. You seem humble, patient and take your time covering details from more than one perspective and so make it easier for those of us who struggle with mathematical concepts and processes to follow your presentation. Thanks!

[@julianbarrett3561]

Wow! The way you explained this made a lot more sense to me than my professor's explanation and the other videos, especially the negative values shifting to the right and the positive values shifting to the left. I had trouble wrapping my brain around the reason until now. I'm taking engineering and am required to take calculus 3, but I can't get there without understanding pre-calculus. So I really needed this video.

[@halicusdiaarcan102]

i hope you got there!! best of luck to you in engineering :)

[@georgesadler7830]

Professor Leonard, thank you for another awesome introduction to Graphs Transformations. The library functions really help me too fully understand all levels of Transformations. This type of teaching style is off the learning charts.

[@goldensilver793]

Thanks Doc... Really appreciate the time you put into these videos you made for us...This gift you gave us...

[@ilgizdavidov3888]

Thank you Professor Leonard!!! So awesome to spend time with you.

[@ammishtandon8267]

I studied calculus 3 from your videos and my exam went great this semester. I'll be taking differential equations next semester starting from january and i want to study from your lectures so please cover the leftover topics like second order DEs, Laplace, Wronskian, Dirac's Delta function ,eigenvalues and all. I beg you please..

[@PastryDonut]

I think horizontal shift notation can be easier to understand if in "f(x+h)" you replace "x" with any other letter such as "w", because those x's represent different things. Then "f(x+h)" becomes "f(w+h)", where "w" is your horizontally shifted input value. Your input must remain equal to f(x), because it's only horizontal shift. Therefore, f(w+h) must be equal to f(x). To find "w" drop functions and solve: w+h=x --> w=x-h. So, shifted input "w" is less than "x", hence shift left.

[@niki2007-y1p]

More than the educational content and explanations, my learning quality depends on the vibe i get from the teacher. I felt realy relaxed and confident while listening to you which doesn't happen often! I hope the positive energy and my gratitude reaches you:)

[@mamadkhan6708]

The best channel I've ever seen on youtube

[@irace5206]

God Bless Leonard the Bro is a life saver. I love that he breaks it barney style

[@thatyoutubeaddict]

Your explanation has made this lesson much more simple,thank u so much professor❤

[@nathangraber8447]

The timeline analogy was a nice little addition

[@thirumalreddy3134]

Your explanations are pure gold! Grateful for your channel

[@pamelagonzalez7684]

Oh my WORD!! Thank you so very much for your lectures. I was in the middle of getting frustrated, and staying behind in my math. Took me over a week to get through these concepts in my book. Then, your videos popped up! You were super "Eye Candy," so I stopped and said, "PRRR." Most importantly, I FINALLY comprehended in 48 minutes what took me over two weeks!!! Thank you thank you!! Blessings to you and yours. Keep on doing what you're doing.

[@socialonly4491]

Hats off Professore, you beautifully explain the concepts again

[@yashaswikulshreshtha1588]

As far as i understand the logic behind (x- k) is that you need to do that while not affecting the Y value. In function if you affect x then it affects y as value, the question is how do you affect x without affecting y and that's where the logic of opposite things comes in.

[@mattdrafted2024]

Thank you Professor Leonard for another great math video. Thank you !!!!!

[@Ckdude100]

Professor if I become rich in the future (big if obviously) you're going to get a very large package of money

[@ProfessorLeonard]

I appreciate the sentiment :)

[@pullingthestrings5233]

Are you still broke?

[@lyricass7810]

Dude are you rich now ?

[@Peakcellfr]

​@@lyricass7810It's been 4 years he either just graduated or still in uni so yes he's still broke

[@niki2007-y1p]

Update us please

[@culture3469]

Happy Teachers Day Professor Leonard ❤

[@weekimotion]

You are Lucky if you are watching this believe me ur on right track

[@andrewesquibel9614]

Another way to show why f(x) = |x+2| is left-shifted: invert the graph, solve for x. The equation becomes x = f(x) - 2 or x = y - 2. Shifts intuitively, just like y = x +/- k.

[@horrorpretzel]

Wow I’m glad you posted this. I have a test on this tomorrow

[@alambeaux8679]

thanks for the great video king, hope you and your family are doing well

[@nurqadwa2223]

Struggling to complete my hww, thnk you sm for thisss

[@UnitedGMV]

woah the camera quality is awesome in these ones i'm going off those older calc 1 videos from like 2014 just came to see if you were still buff... got my answer ;P

[@hkanhai5094]

Hi professor Leonard.. I know u are a busy person but I am patiently waiting for the rest of the differential equations.

[@nm1978]

my god, what an awesome video and explanation. you are the best on you tube.

[@blu3__skies679]

Missed a golden opportunity for "when we get to trigonometry it's gonna be a piece of Pi to graph them", instead of "piece of cake" haha

[@AmitKumar-os8xq]

negatives inside square root is not defined for real functions but you have drawn its graph for explaining reflections. hows that Professor ??

[@edvincejoymedesdannug1142]

I think that's because there's already a negative sign inside the function? Like... y=f(-x) is f(x)=√-x, then let x= -x, then you'll have: f(x) = √-(-x) that would make it positive: f(x)=√x. Hope it's clear. I'm not really good at explaining hahaha.

[@ethioupdatedinfo.]

Thank you very much, Professor Leonard. You helped me a lot.

[@rogue_gamer2490]

after completing my bachelors , i wish to meet you or even once as long as im alive . lots of love from pakistan.

[@nathanborak2172]

What you say at 32:03 is wrong. A vertical stretch is not always a horizontal compression. This is only the case for functions with the property that f(ax) = bf(x) for some a and b. Most functions do not have that property!

[@FethiMuktar]

Thak u 😊 u can use interception as center of transformation

[@GeorgiosMichalopoulos]

Simply amazing!! Thank you for uploading thisseries.

[@somasahu1234]

Awsm explanation!

[@DrMasi]

Good job professor

[@LudicrousValiant]

Here is a mnemonic for the (x+h) // (x-h) distinction in shifting left or right: "To plus in P's ain't right to be, instead to minus is righteous."

[@cvm-xi-2vijayatharishans167]

@14:29 Watching this in june 21, 2023. If you are watching this in future (for me future not for you),

[@zubeirabdulqadir3719]

THE GOAT OF MATH💎💎💎💎💎

[@djt6fan]

Prof. Leonard how many vids do you plan on uploading in this playlist?

[@justinbodden2682]

Professor Leonard: im 6 feet tall. Me: u aint have to flex on me like that....

[@ichigokurosaki7762]

Can you do linear algebra or an upper division course please?

[@Peter_1986]

I have honestly always kinda wondered when Professor Leonard was born. My guess is that he was probably born some time in 1978-1980, given his clues such as being a teacher for 13 years in the year 2019 and apparently being more or less in his early 30s during his Statistics videos from 2011 and things like that.

[@yourdailydoseofreality3219]

I don't understand the last example. How could a negative number be put under the square root?

[@trajansanchez5075]

You are a lifesaver!

[@ericsong19988]

my exam is in 2 days can you please do fourier analysis (differentiation/ integration) Regardless I love your content and thank you for everything.

[@SashahWilson]

you are a life saver

[@Dontworryboutit247]

PLEASE DO A SERIES ON LINEAR ALGEBRA. I'm literally failing this thing... :/ *cries*

[@IndaneGasStove]

Ali Syed try 3blue1brown for now

[@nm1978]

can you show us solutions to equations like ax3+bx2+cx+d=0 and higher roots? I like to see these.

[@clinkclink7814]

12:09 Love that word play. Witty as ever.

[@فففف-ج6ح]

Thank you very much this was very helpful

[@aleksandarjankovski6542]

Freaking great explanation. Bravo!

[@abhishektayal7549]

In the last graph, If we change the input sign. then , the input 1 became -1. but, It can not be possible prof. Square roots don't have negative inside them. Please clear my Doubt Professor.

[@cizzle2054]

Man you need to get your shift straight. Best line from a teacher ever lmao awesome

[@jump7holes]

Thank you sir! Keep going

[@sellakc]

So if you are given a translated graph, are there more than one solution? For example, I have an original function f(x)=*(1/2)^x. The shown graph crosses x=0 at y=3. and the asymptote (normally at y=0) is shifted to y=1. I get y=(1/2)^(x-1)+1. The "correct" answer is 2(1/2)^x +1. These two graphs are the exact same when plotted. Is one answer right and one wrong? Or are both okay?

[@philosophertiger4124]

Really great video thankyou so much,

[@amymccann8760]

"You need to get your shift straight",LOL 😂

[@SmileKaroYaar]

I NEED YOUR HELP WITH LAPLACE AND FOURIER TRANSFORM PLEASE, SUGGEST ME SOME BOOKS OR CHANNEL'S. Q- ARE YOU GOING TO COVER THEM NEXT YEAR?? THANK YOU🙏

[@kufreibanga7980]

The last example I guess was wrong cuz u cant have a negative in a square root which will give a complex number.

[@christianisaac2816]

Not sure if you still check comments. Back a couple videos you discussed the inability to use negative inputs in even roots. This being said, for reflection about the y-axis, i.e. y=f(-x) as in your last example shouldn't it stand that you are not able to reflect even root functions given you are changing the sign of x prior to utilizing the operator which as discussed before cannot operate on negative numbers. And if all your inputs are already negative, yes it could work, but that would've been an imaginary function you were given because it wouldn't operate on negative numbers to begin with so seems kind of redundant. Love you to death but if someone can elaborate please do. I get the general idea of y=f(-x) and the reflection about the y-axis I just think maybe it's functionality when these even roots appear has been overlooked unless I'm missing something.

[@prashantsingh3664]

same doubt bro

[@NYCGemini2000]

for the last graph, how are you allowed to change the input of x to a negative? isn't a negative undefined and you wouldn't be able to graph it?

[@michapau7354]

you can only put in negative values (and zero) for x (which than become positive -(-) = +)

[@lauhityaaashwinkumar]

Thank you so much professor❤

[@dalekelley800]

12:08 ha ha, I like the real SHIFT he is talking about!

[@pullingthestrings5233]

12:09 "man you need to get your shit straight" 🤣

[@marktravelinPI]

can some one explain to me: 45:46, how can 1 be in input to sqrt -x. i am not understanding the concept of that. or maybe im just a bit lost. i understand how -1 can be an input, but 1 seems like it would end up being the sqrt of -1.

[@rosesarerosie9946]

14:20 his example was zero so he needed to plug in -2 but what if, idk, we need 4? we would plug positive two? idgi

[@NoOne-jo9qv]

Brilliant, thank you

[@ppmp9937]

In an absolute value graph: your horizontal shift and Y-intercept have the same values. right?

[@wyattp1668]

"You need to get your shift straight" 12:10

[@sastavideoswala]

Waiting for linear algebra 😊

[@AndrewTota]

how do i know if it is even or odd my teacher explains ut as positive or negative

[@feilongish]

You guys gotta get your shift together😂😂

[@captainsunchipz2521]

Get your shift straight! Love the dad joke hahahah 12:03

[@thebetterewing444]

For vertical shift/ compression, what if the number is equal to one?

[@thebetterewing444]

Oh wait, dumb question.......🤣

[@SMCLabs]

Thank you sir