Graphing Sine and Cosine Functions with Transformations (Multiple Examples)

Learn how to graph sin and cos in this video math tutorial by Mario's Math Tutoring. We go through 7 examples as well as show how to graph the basic sine and cosine graph using the unit circle. We discuss the transformations such as the phase shift, vertical shift, amplitude and period as well as when the graph is reflected over the x-axis. We discuss how to find the period and how to find the scale for your graph.
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• Graphing Sine, Cosine,...
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[@mark8877]

I’m a professional full-time math tutor, and without any caveats I can say that Mario’s tutorial videos are some of the best on KZbin. No one is as clear, concise, or thorough in their explanations as Mario. I often recommend/send his videos to my students. The only thing I have against this video is that the period of a sin or cos function is 2pi divided by the absolute value of b, rather than just b. Regardless, this did not matter for this particular video because all b values were positive, and overall it’s still an excellent video and explanation.

[@zambo8610]

Hi, in a lot of coarses (mine as well) is b defined as a strictly postitve number. So I think that he also follows that rule

[@rebeccasmerdal7784]

I 100% agree! I am a full-time professional Math tutor as well, and refer my students to this guy! The videos are on point, and I like how he provides lots of different examples for students to gather a full bodied understanding from 👌

[@ninadee9753]

Hi! Can you explain why the period = 2i/b? I am so lost with that

[@lukebowman9260]

@@ninadee9753 the period of the parent functions of sin and cos is 2pi. So, in order to find the new period you have to divide 2pi by the number in front of the x value.

[@Hanin-u2r]

Are you sure😅

[@rottenstump0]

I felt very overwhelmed by this in my trig class but your videos saved the day! I got a 38/40 on the quiz for graphing trig functions. Seriously, thank you so much Mario! Your videos are excellent.

[@MariosMathTutoring]

Awesome! Glad my video helped you!

[@niallspotat0e823]

i think your method of teaching how to graph considering the phase shift and period is the best one I've seen. that whole thing about making your new origin really helps

[@DanielA-pg2no]

Genuinely saving my grade I love you Mario

[@MariosMathTutoring]

Glad my video helped you Daniel!

[@georgeh1352]

Had to learn advanced Trigs much later in life, this video has filled in some gaps of my knowledge. I'm 75yrs old, so still learning out of curiousity and keeping the mind alert. Making the concept of a 'new origin' is the 'penny dropper', I guess that's the difference between a Teacher that can teach a subject and one that can't. Many thanks for your video.

[@MariosMathTutoring]

You’re welcome!

[@jasonwolters7306]

This man is the GOAT. If my teacher could explain this 1/2 as well as you could I would have an A+ in the class

[@generalmadness6832]

Same here, my teacher is awful

[@amahimahiaha]

Thank you for the variety of examples! The last one is the one I've been stuck on and was explained so easily and clearly.

[@xasunaa]

Illustrated it perfectly! In our functions class the teacher never touched over the period and how dividing it by 4 gives you the quarters, which was crucial to my understanding of the sine & cosine waves back in grade 11. Thank you for the clear and concise video ^^

[@MariosMathTutoring]

Awesome to hear it helped! It's important to know how to break down the graph and see the key parts.

[@gutierrezjeffryv.7649]

you're such a really great professor I have ever found here in youtube. Your explanation is well-delivered and comprehensible.

[@MariosMathTutoring]

Thanks! 😃

[@hellohudsons]

Thanks!

[@MariosMathTutoring]

Thanks for the Super Thanks!

[@chandra68]

One of the best videos on trigonometric transformation. Concept and problem solving were taught in a simple yet effective fashion. Hats off to your teaching abilities. Thankyou and weldone.

[@sweethmv]

I wish that I had teachers like you in school. Your voice is calming. Your delivery of information is smooth and simple. Thanks. You have made college for me possible.

[@MariosMathTutoring]

You're very welcome!

[@Sumire1107]

wow, really made it easy for me, i was struggling with this until i just watched this video. thanks a ton mario, i really appreciate the work youve done to help people like me

[@MariosMathTutoring]

You're very welcome! Glad to hear my videos are helping you.

[@sarafinamcinnis9990]

thank you so much mario. i've been trying to graph this one sin one for hours and I finally got it because of you!

[@ddahstan6876]

Super duper easy to understand and follow. Love the Period/4 to nail down the graph at the get-go... and the rest can pursue to the rhythm.

[@maria-zn7jd]

u saved me ive been crying for 5 mins. i watched this video now i now fully get it. thank you 🙏

[@MariosMathTutoring]

I’m glad my video helped you understand the concept of transforming sine and cosine functions!

[@Op_Hunter5]

Thank you mario.I really appreciate your mathematics technique with us.

[@MariosMathTutoring]

Glad you like them!

[@andrehardaman9375]

You have no idea how much this helped me, I was so lost before and this helped me understand and complete my homework assignment. Thank you so much.

[@PaulGovcorrupt]

I am a 55 YO, Pre calc student. Lol (Stay teachable)This has really helped understand this topic and these formulas. Thanks

[@MariosMathTutoring]

Glad it was helpful!

[@PaulGovcorrupt]

Are you for hire on a 1 on 1 basis ?@@MariosMathTutoring

[@RaffaelloLorenzusSayde]

To clear the confusion for some people: We are looking for the period in 1 quadrant in a total of 4 quadrants (horizontally) of a unit circle (portrayed unto a sinusoidal graph), so that would be the equation 1/4 quadrants times the period 2pie/|b| = per 1/4rth of a period. The starting point will be d and thus, we will add by the 1/4rth of a period each time. Also, he might mutliply the denominator by common multiples of the 1/4rth period and the starting point d to get the LCD of both matching denominators (only if one or both are fractions). Btw, every curve is a quadrant.

[@durdonabakhtiyor2534]

Wow, it was a fantastic explanation Sir, thank you a lot. Why I didn't watch it earlier. I have watched so many tutorials on the KZbin channel but yours is one of the best which I have ever seen 👍👍👍👍

[@yojacq]

God bless you man this is so much easy to understand and follow

[@rodneysteenekamp8924]

I was really battling with this. Your video helped me alot, the work makes much more sense to me now. thank you!

[@ahmadjamal2320]

One of the best videos I have seen so far. Keep up the good work mate.

[@Brie.açaí]

This is better than my school’s actual tutoring center.

[@banuner_]

Doing it this way makes waaaaaaay more sense to me

[@MariosMathTutoring]

Awesome! Glad the video helped make sense of graphing sine and cosine graphs.

[@alexispoluvatzick6163]

Thank you for saving me for my pre calc midterm tmr ❤️

[@MariosMathTutoring]

Good luck!!

[@rkumaresh]

Thanks so much. You have cleared most of my doubts in this topic of graphing the sine and cosine functions. I could watch it as many times as and when I require them. Regards Tutor Kumaresh

[@Aest0pulence]

4:44 9:07 11:20

[@generalmadness6832]

way better math teacher than the one i have.

[@阿额]

Fr

[@user-yt6hw2yb7p]

I have an exam and you saved my time with your perfect explanation. Thank you

[@bluecat7431]

Thank you so much! This was one of the topics I was really struggling on.

[@jonahheschel1199]

Thank you!!! This helped me so much since I missed the start of this unit.

[@MariosMathTutoring]

Glad I could help!

[@hellohudsons]

Wonderful videos. I’m studying for my Precalc 2 final, and your videos have been a great help. Thank you!

[@MariosMathTutoring]

Awesome!

[@justin-en]

That's a really awesome tip at 3:12 Thank you!

[@motoverse-k7z]

Thank you Mario ❤

[@metookwok6545]

thank you sir ! you save my math and my life🥲🥲🥲

[@yinyang9508]

He explained in 14 minutes what my professor takes 2 weeks.

[@dionpaulcreations960]

life saver. Liked, subscribed, commented.

[@MariosMathTutoring]

Thanks for the sub!

[@irwi_n4078]

This guy is honestly goated

[@ferozdba]

thank you very much for the clear explanation with examples. it was very helpful.

[@MariosMathTutoring]

Glad it was helpful!

[@soybean_TM]

Thank you so much Mario! I found this video very helpful

[@jaimiewallace3127]

you just saved my life

[@curtpiazza1688]

Excellent presentation...I wish I had you 40 years ago!

[@gracepark3188]

You are my savior. I bow down hallelujah.

[@jamesanterola3680]

!!!

[@GiftMakumbilo]

sir you the best no lie

[@thytran271]

omg u save my life ❤😭

[@theman7050]

This is brialliant and exactly what I was looking to understand for generative art. Thank you

[@ninadee9753]

11:04 one thing I do not understand about this is the fact that when you plug in 2 for the equation for #6 it doesn't equal 0?

[@sofiyanoroozi2021]

You saved me from failing ❤️😁

[@yasmineyasmine8059]

Thank you very much, you saved my grades!!!

[@بدوننام-خ7ص]

That was extremely useful

[@MariosMathTutoring]

Glad it helped

[@isabellawilliams52]

god bless you mario. I hope your pillow is COLD tonight.

[@Soltani88]

This saved the day! My teacher did a horrible job teaching this compared to you. Thanks

[@arslanmoghal5472]

Very well explained.. Good video 👍👍, thank you

[@captainhoops9922]

thank you very much this was very helpful

[@MariosMathTutoring]

Glad it helped

[@nicholasmichaelrusso6924]

best video ever

[@rojanbowl5064]

Thank you. God bless you!

[@thelight1622]

Amazing explanation

[@Rina0027]

What will be the range of example 7 ?

[@tolinhnguyen79]

Thank you very much

[@rafaelibarra280]

Very clear. Thank you

[@MariosMathTutoring]

Glad it was helpful!

[@ryanlorenti469]

you’re an amazing teacher bro stg

[@yassinsohayly8362]

thank you dude

[@josefafox4908]

Thank you so much!

[@MariosMathTutoring]

You're welcome!

[@AngelMae-e3h]

when graphing should i stop at the first period or i should continue?

[@MariosMathTutoring]

Depends on what the instructions state….sometimes it will say graph from -2pi to 2pi or it might say graph 2 full periods, etc

[@ohhesnumb4409]

Well explained

[@adityasingh1058]

amazing video sir

[@attusiii]

I love this human

[@MrBej]

Very helpful, thanks!

[@MariosMathTutoring]

You’re welcome Nayem!

[@qn9mt]

THX SO MUCH YOUR SO MUCH BETTER THEN IXL

[@MariosMathTutoring]

Thanks! Glad my video helped you!

[@jakahifreeman6614]

you’re the best at explaining things helped a lot thank you!

[@MrBeafyYT]

thank you so much for this video, helped me out a lot!

[@phwataabdul8272]

are you master Ying C? anyway, your tutorials are best

[@arifhossen7785]

From Bangladesh seeing...Thanks

[@sandwich9404]

Hey Mario, What math course is this?

[@MariosMathTutoring]

Hi Tyler...usually Algebra 2 and/or PreCalculus...covered as the trig part of the course.

[@aidanw6725]

Thank you

[@johnmicoong8106]

Thank you this is very helpful..

[@WillPak]

I think this video is great! However, I believe the way you represented the formula could be taken incorrectly. A sin b(x-h), i believe it would be better to represent it as A sin (bx-h) + . I got mistaken at first and i would distribute the b between x and h to get bx-bh. I understand why you did it, because it's easy to see and distinguish what b is when they are trying to find the period, however I believe it could be taken incorrectly. I could also be incorrect in my evaluation but I just thought I'd let you know! :)

[@MariosMathTutoring]

This more recent video I did on the topic may help clarify it more for you: Graph Sine and Cosine Graphs Step by Step From Easy to Difficult kzbin.info/www/bejne/mYDbZaydrK1sj7c

[@hollypoole4333]

this was so helpful

[@Youwishyouknewwhohuh_]

will you always divide the period by four? can you explain more why you divided by four

[@RaffaelloLorenzusSayde]

We are looking for the period in 1 quadrant in a total of 4 quadrants (horizontally) of a unit circle (portrayed unto a sinusoidal graph), so that would be the equation 1/4 quadrants times the period 2pie/|b|= per 1/4rth of a period. The starting point will be d and thus, we will add by the 1/4rth of a period each time. Btw, every curve is a quadrant.

[@kwang-suglee7423]

Thank you!

[@MariosMathTutoring]

You’re welcome Kwang-Sug!

[@kaylale2911]

Pr

[@Youngballer739]

at 11:00 The maximum should be 2, not 1

[@tont0665]

you are the best

[@habebawael5099]

thank you 🙏🏻

[@abrarkhair5346]

You are amzing

[@MariosMathTutoring]

Glad you like my videos!

[@waja6001]

Good

[@ElCruncheroEditz]

for number 4 isn't 5pi/4 supposed to be 9pi/4?

[@justinels2632]

could you graph a more complicated equation, such as, y= -2sin2(x+ pie/6) +1, and the domain be [-pie,pie], i cannot seem to find something like this on the internet, i struggle with identifying the endpoints of this. thank you

[@MariosMathTutoring]

Justin, this other video I did on graphing trig functions may help you: Graphing Sine, Cosine, Cosecant, Secant, Tangent & Cotangent (Complete Guide) kzbin.info/www/bejne/Y5_VZn55ntmXfdk

[@jeyakumari]

but why are dividing the period by 4 in both sine and cosine graph?

[@slovajleclerc9093]

Because circle dividend into 4 parts through a full cycle

[@RaffaelloLorenzusSayde]

We are looking for the period in 1 quadrant in a total of 4 quadrants (horizontally) of a unit circle (portrayed unto a sinusoidal graph), so that would be the equation 1/4 quadrants times the period 2pie/|b|= per 1/4rth of a period. The starting point will be d and thus, we will add by the 1/4rth of a period each time. Btw, every curve is a quadrant.

[@ninadee9753]

Why does 2pi/b equal a period? I've thought that is so arbitrary

[@rypto8700]

i love you

[@Algebrainiac]

These take so long to graph. They’re very very hard for me…. I’m a math tutor and I make math vids so I haven’t made videos on this. They’re a b*tch to do…!💀💀💀

[@icebreakmusic2291]

what? still searching KZbin for math help? are the videos from 2006 really helping? yes, yes they are, now get your crappy PowToon lookin animations outta here.

[@suza_627]

I am so confused. I'm gonna fail.

[@RaffaelloLorenzusSayde]

If you're confused about the period. Here is my explanation (hope it helps): We are looking for the period in 1 quadrant in a total of 4 quadrants (horizontally) of a unit circle (portrayed unto a sinusoidal graph), so that would be the equation 1/4 quadrants times the period 2pie/|b|= per 1/4rth of a period. The starting point will be d and thus, we will add by the 1/4rth of a period each time. Btw, every curve is a quadrant.

[@may8128]

did you fail?

[@dumbwithmilan]

imma fail.😢

[@zoro.mf69]

Sir thank you so much

[@dbuc4671]

I’m a professional full-time math tutor, and without any caveats I can say that Mario’s tutorial videos are some of the best on KZbin. No one is as clear, concise, or thorough in their explanations as Mario. I often recommend/send his videos to my students. The only thing I have against this video is that the period of a sin or cos function is 2pi divided by the absolute value of b, rather than just b. Regardless, this did not matter for this particular video because all b values were positive, and overall it’s still an excellent video and explanation.

[@sunshineli9949]

stolen comment

[@nnizzel1003]

Thank you!!!

[@dbuc4671]

To clear the confusion for some people: We are looking for the period in 1 quadrant in a total of 4 quadrants (horizontally) of a unit circle (portrayed unto a sinusoidal graph), so that would be the equation 1/4 quadrants times the period 2pie/|b| = per 1/4rth of a period. The starting point will be d and thus, we will add by the 1/4rth of a period each time. Also, he might mutliply the denominator by common multiples of the 1/4rth period and the starting point d to get the LCD of both matching denominators (only if one or both are fractions). Btw, every curve is a quadrant.