Introduction to Graph Transformations (Precalculus - College Algebra 14)
Professor Leonard
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How to use Transformations to Graph basic functions and why the transformations do what they do.
20 Comments
@rooh-ezehra3062 3 weeks ago
i thought that finally college was when my downfall finally started but turns out my teacher doesn't know how to teach
@anima9696 1 year ago (edited)
Professor Leonard is 6 feet tall, looks like superman, is buff, and can teach advanced math like a god. WHAT IS THERE THAT THIS MAN CAN'T DO???
@angelina_assis 4 years ago
I am in high school in Russia. you explained the topic in 40 minutes better than my teacher in a week. Thank you
@jaime9331 8 days ago
I hope my comment helps out if the Horizontal Shift still confuses you.
what it helped me out was
x=0
this we can call it the origin of our function
x+2 = 0
x=-2
this we can call it our adding units to our function.
so every time we want to add values to our function we have to move to the left.
x-2 = 0
x =-2
In order to be able to subtract units from the origin of our function we need to move to the right .
I used Claude to better phrase it so it came out like this:
"A horizontal shift moves a graph left or right. It happens when a number is added to or subtracted from x inside the function, as in f(x + h) or f(x − h).
To find where the key point moves, set the inside equal to zero:
f(x + 2): x + 2 = 0, so x = −2. The graph shifts 2 units left.
f(x − 2): x − 2 = 0, so x = 2. The graph shifts 2 units right.
The shift goes in the opposite direction of the sign: adding inside moves the graph left, and subtracting inside moves it right."
Example to include: f(x) = x² has its vertex at x = 0.
(x + 2)² has its vertex at x = −2, shifted left
(x − 2)² has its vertex at x = 2, shifted right
Quick check: For (x − 2)², the vertex is where the output is 0. Plug in x = 2: (2 − 2)² = 0 ✓. So the graph moved right to 2."
Good luck to you all watching the video, and Thanks professor Leonard for the channel.
what it helped me out was
x=0
this we can call it the origin of our function
x+2 = 0
x=-2
this we can call it our adding units to our function.
so every time we want to add values to our function we have to move to the left.
x-2 = 0
x =-2
In order to be able to subtract units from the origin of our function we need to move to the right .
I used Claude to better phrase it so it came out like this:
"A horizontal shift moves a graph left or right. It happens when a number is added to or subtracted from x inside the function, as in f(x + h) or f(x − h).
To find where the key point moves, set the inside equal to zero:
f(x + 2): x + 2 = 0, so x = −2. The graph shifts 2 units left.
f(x − 2): x − 2 = 0, so x = 2. The graph shifts 2 units right.
The shift goes in the opposite direction of the sign: adding inside moves the graph left, and subtracting inside moves it right."
Example to include: f(x) = x² has its vertex at x = 0.
(x + 2)² has its vertex at x = −2, shifted left
(x − 2)² has its vertex at x = 2, shifted right
Quick check: For (x − 2)², the vertex is where the output is 0. Plug in x = 2: (2 − 2)² = 0 ✓. So the graph moved right to 2."
Good luck to you all watching the video, and Thanks professor Leonard for the channel.
@amirbassiri1965 1 year ago
the moment at 14:39 when you say let's say that the years 2015... and it's 2025 and I am watching your lectures...! guys be careful please, time flies sooo fastttt, so do your best like THERE IS NO TOMORROW, because I was watching professor leonard lectures in 2021 and it feels yesterday so always think about it that time flies and do your best like THERE IS NO TOMORROW
@TheDemarie76 4 years ago
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.
@ammishtandon4961 6 years ago (edited)
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!
@1Vice876 6 years ago
I rememer watching Prof. Leonard videos two years ago when I was talking Calculus. Thank you🇯🇲🇯🇲🇯🇲
@MahdiSadeghi-ly2th 2 years ago
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.
@ysmko 8 months ago
wow i'm shocked. i was struggling for weeks with understanding the foundations because i just wanted to pass before going to calc 2, then the topics from there slapped me in the face. You truly decreased my mental anxiety. Thank you so much.
@JebusXLT 10 days ago
Watching this and your other playlists for review (currently in calc 2) in Fall of 2026. Thank you for your now 21 years of an exceptional job teaching!
@Ckdude100 6 years ago
Professor if I become rich in the future (big if obviously) you're going to get a very large package of money
@skippyelder 5 years ago
It's amazing how you turn something that I always saw as complicated and you explain it in a simple yet completely accurate way.
Introduction to Graph Transformations (Precalculus - College Algebra 14)
Professor Leonard
231,544 views
Support: Patreon: ProfessorLeonard
Cool Mathy Merch: https://professor-leonard.myshopify.com
How to use Transformations to Graph basic functions and why the transformations do what they do.
The content and the simplified explanations are still helping the many students across the globe to study the advance concepts of the mathematics in the easiest way.
Thanks Sir
Student from India 🇮🇳🇮🇳🇮🇳