Transforming Algebraic Functions: Shifting, Stretching, and Reflecting

Now that we know the basics regarding graphing algebraic functions, it's time to learn some tricks that will come in handy as we graph different kinds of functions. Specifically, there are terms inside functions that do very predictable things to graphs, and if we learn what these are, we can look at certain functions and quickly graph them without making any tables, even functions that look super difficult. Who doesn't like a shortcut? Check this out!
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[@mynamesimm]

I've been struggling to understand graphs all semester and you just explained it in 7 minutes, I actually love you

[@owenegona2632]

fr bruh

[@alibinnaseer]

Brain.exe crashed, watched 4 vids today of your mathematics series. The highest i watched in one day. I have watched 65 vids so far. You are the one to be called a teacher. You have earned my respect sir. May Allah bless you!

[@pikleman5880]

Same.

[@Stoirelius]

I watch 10 per day.

[@ChefSporty]

@@Stoirelius you are not him

[@djpriest123]

bro does not have the apple emojis@@ChefSporty

[@ChefSporty]

@@djpriest123 i have the iphone 42

[@ercm2393]

Ur a beast...much better at explaining this concept than my teacher.

[@lastpot261]

Eric Moore any1 is better than my teacher, he doesn’t even try to teach he just hands us the notes and is like alright here is your homework and then does nothing for the rest of class

[@lawrenceliang6670]

@@lastpot261 reason for that probably stems from public school teacher's confidence in their job security, but KZbinrs have to lecture well to attract more popularity? But in either way this KZbinr is really well at lecturing so I'd rather to spend time with him even though he charges in the future.

[@basketguitar9768]

​@@lawrenceliang6670hi

[@yotelolailo]

In the final transformations table the horizontal shift is the other way around as stated during the video and the final comprehension exercises. A positive h means a shift to the left and a negative h a shift to the right!

[@mathconcepts101]

saw that too, i had to watch the video again

[@thecrazyguy3053]

Thank you so much but you made a mistake in the summary of transformation for the h variable. Positive h= shift left, negative h= shift right . You’re the best❤

[@ProfessorDaveExplains]

The table is correct.

[@geniusvadim7858]

@@ProfessorDaveExplains Isn't it supposed to be opposite, where -h = shift right = positive?

[@denzelzuofa3066]

@@ProfessorDaveExplains You are wrong, but it doesn't really matter.

[@poopyhead7586]

@@denzelzuofa3066 (x - h) here h is unknown so if we put any negative no. it will be ( x - (-h) ) making h positive so it'll shift left. :) hope i helped.

[@curtisdammann5043]

I just finished my first night class for Pre-Calc30 and this video made everything we covered last night make a lot more sense. I really appreciate your efforts. I will watch more of your content as I am also upgrading my CHEM 30

[@Kattegat08]

I'm in college algebra and I was seriously struggling with this. I did not get a good grasp from my professor. I watched videos from my online textbook, khan academy, and multiple random vids on youtube before I found yours. Thanks so much for the concise explanation!

[@basketguitar9768]

Did u graduate yet

[@ikilledaman]

Thank you very much, professor dave. I’m watching a bunch of these videos to help me with my Algebra 1 class, as my teacher (and most US schools tbh) just teach “this is the formula, solve now” and not the actual concept in a way that’s easy to understand. Your videos make it a whole lot easier to grasp the concept of what all the x^2 stuff actually is doing to the function, and even made clearer what a function *is* . Thanks

[@rysterstech]

So sad yet so true..................................

[@jumakhanrahyab6666]

In the table, for horizontal shift-- positive h = shift to the lift instead of the right. Am I right?

[@AriaHarmony]

Very simple and to the point. Thank you very much!

[@chad8519]

im confused about the horizontal shift transformation on 6:28. I understood that (x+h) shifts left while (x-h) shifts right on the graph, or something I missed?

[@llrree778899]

That's correct. Think of the minus sign in (x-h) being part of the original form (like a formula). If h is positive, you have (x-h). If h is negative, you have (x-(-h)), which becomes (x+h).

[@athalianicole5081]

It seems like the summary has some unintended mistake. You said that if h is negative, it should be shifted to right. But on the summary, it says that if h is negative, it should also shift to left. I'm confused, professor.

[@ProfessorDaveExplains]

but don't forget that minus a negative h is positive

[@sheISomuwa]

An absolute genius! He deserves an AWARD!

[@kristinbotley7896]

The whole video is summed up at 6:32

[@pppp_627_u]

thanks

[@atullizard5219]

Thank you for this Just one doubt What is the difference between vertical and horizontal stretch?

[@vaishanthjv2519]

Same doubt

[@ceegae1670]

If you vertically compress it would look like a bowl, and if you horizontally stretch it would look like a huge U

[@slickyy21]

This is getting overwhelming right now Can't remember this much of concepts.

[@farhadjoya7882]

I have seen many videos about this subject. Your explanation is the best ever. Thanks

[@Marinameyers05]

Okay, can someone please explain the difference between the horizontal and vertical thingy for Dilation.

[@Kawamata123]

Good video. I've read the order of operations matter when transforming functions. First we do the innermost parenthesis (horizontal shift) , then the horizontal/vertical stretches, and then the vertical shift. f(x) = -2(x - 3) ^2 +4 worked the way you did it, but there are cases where it won't.

[@anamikasingh5550]

I can’t thank you enough for simple methods

[@ilovecherriesa4860]

He is so much better than my Algebra teacher! Thanks for this lesson 🙂

[@anganileamon7813]

Prof Dave , you are talented

[@motherof3pearls]

I so love this! On my way to understanding things! Thank you!! 🤗

[@yirandai681]

Is it "horizontal compress" at 3:37 instead of "horizontal stretch" as you say? Please check it up.

[@ProfessorDaveExplains]

a stretch can be applied in the positive or negative direction, either way we refer to it as stretch.

[@mathacademyforall4161]

@@ProfessorDaveExplains it either compressed or stretched ?????

[@UthixunamHewu__]

Hi, please correct the summary table: Horizontal shift part where negative h= is supposed to be to the right and positive h= shift left

[@Ame-jt2yf]

I can say that you and brian are really good at teaching, straight forward, you guys are the best

[@ADwaMoviesandMusicsproduction]

Very helpful video but on 3:39 you said horizontal stretch but you also said if it's better than 1 it's horizontal compress. Just check that keep up.

[@jasonx2539]

I think you should always mention the order of transformations the same way you would do order of operations or the transformation could become messed up if you were to manually calculate it.

[@elhadjitalla3054]

Hello Professor Dave! Thank you so much for these valuable lessons! I have a question about the table @6:34. Isn't it supposed to read "positive h=shift left, negative h= shift right? Is there a typo or did I misunderstand something? Thanks again!

[@doctorcommon6195]

This is what I though too

[@ArthasFB]

It's not a typo. It's because it says x - h. The sign is not included (if it was, it would say x + h). That means that a positive h, which is then inverted, produces a shift to the right. Hopefully that was clear.

[@elhadjitalla3054]

@@ArthasFB Thank you for your explanation. However, I'm afraid I still don't understand because the formula on the table doesn't match the explanation Prof. Dave gives @2:27. I'd appreciate any further clarification. Thanks again!

[@ArthasFB]

@@elhadjitalla3054 The video says x + 2 generates a shift to the left. That is, x plus a positive number. The same way, x - 2 generates a shift to the right. That is, x plus a negative number (or minus a positive number). You can express that as x + h where if h is positive, it shifts to the left and if h is negative, it shifts to the right. In the example, h is 2. But it is easier to remember if h being positive generates a shift to the right. For that, we can flip the sign of h (x - h). This means if we plug a positive number in h, let's say 3, it gives x - 3 and so the shift is to the right. Notice that the sign is not part of h. For a negative value, - 3, it gives x - (-3) = x + 3 and so it is a shift to the left.

[@elhadjitalla3054]

@@ArthasFB Wow! Thank you so much for your patience and kindness. I think I got it now! So it's all about the sign of the number you plug in for h. Thank you again so much!

[@MegaSquiff]

Very clearly explained! A big thanks to you.

[@SW-nx4jz]

Thanks Professor Dave

[@oliviaaivilo8071]

i've never had problems with this concept but the way my teacher wants us to do it is so unreasonably complicated.

[@jasparevarhart1332]

is it inappropriate to call it 'horizontal compression', instead of stretch? or, is it requisite?

[@thubbard722]

did you find out?

[@mercilessanimations161]

@@thubbard722 are you thomAs

[@thubbard722]

@@mercilessanimations161 no haha

[@bahauddinchishte]

You saved my day, sir! I was struggling to understand this concept for my SAT prep, and search lot of YT tutorial but noc explain it like you. Take love, I am from Bangladesh!

[@sudhadatla5314]

what would the transformation be for square roots such as 3square root x

[@user-dk4sy1xe5m]

oh, professor thanks. Actually I also really liked your videos in Astronomy.

[@bearclaws5671]

Not only super helpful but also has a kickass intro song. Wish I could give a super thumbs up

[@hughjanuz4343]

sir, I'm lost at 4:42, if it's x cubed, then it means 2 x 2 x 2, right? can you sir tell me how did you get the 4, i'm really confused sir. Thank you in advance! :D

[@Hour_of_Mercy]

Thanks! I made a special ‘cheat’ card from 6:28 to help me on transformations with my pre-calculus homework. I figured it didn’t have to be as complicated as many have been making it.

[@gracekgosana9123]

Wow I'm speechless. 👏

[@CrazyChillie]

Wow this was awesome!! All day I was struggling to understand and this was so simple and informative. Thank you’

[@michelletorian]

how do u pick which numbers go under the x in the right column

[@pcz6029]

your videos are awesome, keep the good work bro

[@IsabelRodriguez-nv2ue]

Another super great video!!! Thank you so much for creating these educational videos. They truly help. Thanks !!

[@ohsorandom]

This was mighty helpful, thank you so much!

[@sanjay6052]

Bro you're here every time i need help bro... thank you so much

[@vidyamusic6794]

Ur a great man love ur videos keep uploading

[@bbyd777]

ugh thank u for this!! i need to take a quiz & even with notes the whole concept was still confusing u cleared a lot of it up!

[@Zone_Ranger]

I only watch so that I hear the 'check comprehension' music.

[@user-yr3uj6go8i]

It sounds better when it's sped up by 1.5.

[@liz8041]

actually the conditions for streches are not expressed correctly. for example in 6:30 : a>1 is fine but a

[@saltscynical2418]

thanks a ton your vids help alot especially since ive been doing online school

[@aveinmovil]

Is "f(x)=sqroot(x)" a valid fuction?. for every natural positive numbers besides 0 we will have two positive and negative outputs wich does not meet the "one output requierement". Also it would output imaginary numbers on the negative input side but i guess that wouldnt be a problem. Tahnk you Dave and thank you all the comunity!

[@nytr]

Shouldn't it střech horizontally too in the bottom example? It doesn't make sense to me :D 3:49

[@lydmonkey4593]

Would it be the same if you changed the f(x) part? Like if I have f(x+2)=x^2, would it be moved up? Also I love the outro music :>

[@evanlomioan]

thank you

[@ahnafalnafis]

Thanks man, it was life saving! At 6:30, you told, "+h = shift right, -h = shift left". Shouldn't it be "+h = shift left, -h = shift right"?

[@ArthasFB]

No, because the negative sign is not part of h, so a positive value for h is negated and produces a shift to the right.

[@Mr_NoobMANnNn]

the best teacher ever

[@samwise1990]

6:30 - the positive h = shift right, negative h = shift left. I'm sure I'm missing something but isn't this the other way around?

[@StarsNight123]

Professor Dave, the summary page at the ending of the video is incorrect for the horizontal shift. When it's a positive h, you shift left, not right. And when it's a negative h, you shift right; not left. Please read this guys!

[@ProfessorDaveExplains]

You’re confusing the sign on the term with the sign already present in the binomial. The video is correct.

[@Osiris2134]

@@ProfessorDaveExplains YEAH YOU TELL EM DAVE. SPANK HIM LIKE THE NAUGHTY BOY HE IS AND PUT HIS ASS TO BED 😡 😩

[@o3coe613]

great video ! i have a question, if i have this function f(x) = 2^x + 3 and shifted left 8 units, stretched vertically, by a factor of 6, reflected about the x-axis , and then shifted downward 9 units , what would be the final function ? please help

[@mrphanton9681]

even after all this time this is video is very good

[@alexiashanefernandez8582]

thank you so much for this, such a wholesome video to learn! God Bless!

[@Xardas_]

3:45 it's not a horizontal stretch but it's a horizontal shrink !!

[@damon123jones]

You make it sound so easy as rare in a math teacher

[@forevershampoo]

so positive and negative are opposites and shrink means to get wider and stretch means to get more narrow

[@omargonzalez4455]

This video was really informative...thank you!

[@AlessandroZir]

VERY useful and clear!! many thanks;

[@99graham]

Awesome job professor Dave! Thanks

[@dhanwanth07]

Thank you Professor

[@umeshoaib7314]

Keep it up 🙂

[@kausarkhan2302]

do you have a link to ths powerpoint that I can print out. Your are so helpFUL thanks a bunch.

[@ProfessorDaveExplains]

it's not powerpoint it's a video

[@kausarkhan2302]

Oh so not printable. This was very useful thanks.

[@isaacnewtonstolemyjoy]

🥇

[@lytebluestudios]

I love it here.

[@ethanbarel9446]

The page at 6:33 shouldn’t positive h= shift left not shift right same with nagative h should be shift right

[@ProfessorDaveExplains]

Don't forget that the h term is subtracted.

[@jannatbrishty9416]

Yes

[@tGoldenPhoenix]

Thank you.

[@tGoldenPhoenix]

Done this lesson.

[@Crilic3]

Link to the "other strategies later in the series"?

[@ramyapriya9979]

How is it (-2)^3=8

[@CJ-uy3dx]

My only criticizm about this one: I think you should've used more functions as an example, not just parabola. Perhaps, the sqrt(x) and maybe something else throughout all the transformations. Firstly, it would've demostrated the consistency of the methods as opposed to just ways to manipulate quadratic function graph specifically. Secondly, some of the transformations are confusing when applied to x^2 specifically because they render the same result or results with unobvious difference. For example, I was extremely confused with the stretch example: like, what's the difference between 4x^2 and (2x)^2, for example? Or the example of -x^3 and (-x)^3 was confusing, because you are, like, looking at it and wondering: ain't it the same thing (another example function where the distinction between the transformations would actually render the same result would've been great). Also, from your explanation of shifting along the x-axis, it wasn't clear at all how that would work with other functions (that's where the sqrt(x) example would be perfect). Also in my opinion, sin(x) is one of the best functions to show transformations on: I know that trigonometry hasn't been covered yet, but still, just saying.

[@shikazeevods]

Check his trig classes for trig, the way he laid out his lesson is in order of standardized teaching in the U.S., so he covers algebra 1, geometry, alg 2 etc in that order. As for your question regarding the difference for 4x^2 and(2x)^2, there is no difference, they'd be the same line.

[@user-nz1uo4pd5z]

honestly, I had no better explanation then this, if anyone is taking function or calculus, can we connect to help each other please, and thank you?

[@jianghaoyoutube5017]

Nice video. It helped!

[@jaypoizhan]

When h positive doesn't the function shift left? and when negative to the right? isn't opposite what's on the table in the end?

[@ProfessorDaveExplains]

note that subtracting a negative number gives you a positive number and vice versa

[@jaypoizhan]

@@ProfessorDaveExplains Oh I see, thank you very much for quick response and awesome content!

[@jackscoots9664]

thanks

[@yudhisthir7247]

Great video

[@TheFacial83]

👍👍 Which program did u use? Thanks

[@mashettyshekar1247]

Beauty of math.....😍😍😍

[@dinos4december]

SO GOOD!

[@salmanahmed5659]

Ur super sir... Love from Pakistan ❤❤❤

[@ahmedjahenzaib148]

pakistan zindabad

[@naqeebhussain3357]

Great work u man

[@regachoaliyahkhaet6577]

Thank you

[@thefaustix7560]

you are the best

[@fahadshaikh5686]

Wish i could found your channel more earlier 😊.

[@redyellow5714]

best intro

[@md.shohrabhossainpavel8385]

nice video

[@jacobponferrada597]

okk this is a good video very helpful for my exams the thing is at around 2:14 you said horizontal shit HHAHAHAHAH

[@Osiris2134]

DAVE!!!!!!! I LOVE YOU SO MUCH DADDY!!! YOURE SHIFTING MY FUNCTION UP RIGHT NOW 😏😏

[@holly2327]

My math teacher seduced me to watch this video

[@madhushajosephwimalagaran5160]

Dheivame 🙇

[@jamesbyrne6232]

wouldn't positve h shift left and the other vise versa 6:25

[@ProfessorDaveExplains]

make sure to note that the term is subtracted

[@tomscott9771]

For summary slide at 6:28 horizontal shift(h) shifts are opposite way round +ve is shift left -ve is shift right !! he said this in video but didn't correct it on this slide

[@ProfessorDaveExplains]

the graphic is correct, don't forget to take into account the negative sign before the h term.

[@tomscott9771]

@@ProfessorDaveExplains sorry to be annoying but I think I am still confused, at 2:31 you said f(x)=(x+2) causes a shift left? but at 6:28 it says this causes a shift right ? thanks for your reply

[@ProfessorDaveExplains]

So it's just that you're not considering the subtraction, if h is positive, then you subtract positive h and you get minus h. If h is negative, you subtract negative h and you get plus h. When I'm listing the sign of h, I'm not considering the subtraction sign.

[@tomscott9771]

@@ProfessorDaveExplains okay I understand now thanks for the help !

[@rhanson1124]

@@ProfessorDaveExplains Oh I get it, that is why you specifically noted (x-h), that is the positive h shifting right compared to (x-(-h)) being the negative h which would shift left. Correct?

[@GF86123]

Best Explanation of this ugly concept!

[@gloriasarkar3776]

How -2 square = 4 can you explain

[@ProfessorDaveExplains]

negative signs cancel when you multiply

[@gloriasarkar3776]

But why it doesn’t show on calculator

[@ProfessorDaveExplains]

it should. make sure you use parentheses.

[@gloriasarkar3776]

Yeah now it shows Thanks 🙏