Support: / professorleonard Cool Mathy Merch: professor-leonard.myshopify.com How to use Transformations to Graph basic functions and why the transformations do what they do.
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!
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.
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
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
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.
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!!
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!
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
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.
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!
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.
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 :)
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.
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.
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..
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
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.
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.
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.
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.
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.
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?
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.
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.
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!
