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Nona
12/13/2025
Maths
Transforming Curves
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Transforming Curves
Understanding how curves transform is like learning the secret language of graphs. You'll see how equations change to create different positions and shapes on your coordinate system.
The key to mastering transformations is recognising patterns in equations and linking them to visual changes on the graph. Once you spot these patterns, you can predict transformations without even drawing the curve!
Quick Tip: Think of transformations as giving directions to a graph - "move left 3 units" or "stretch by factor 2" - the maths just codes these instructions!
Complete the Square
Completing the square is your essential first step for spotting transformations. This technique converts messy quadratic expressions into a neat form that reveals exactly how the graph has moved.
Here's the process: take half the coefficient of x, square it, then subtract this value whilst adding the constant. For example, with x² + 8x - 3, you get ² - 19.
The result always follows the pattern (x ± a)² ± b, which immediately shows you the horizontal and vertical shifts of your parabola.
Remember: Half the coefficient, square it, subtract it - this sequence becomes automatic with practice!
Before a Transformation
The basic graph y = x² has its minimum point at the origin (0,0). This is your starting reference point for all transformations.
Tracking how this minimum point moves is the smartest way to understand transformations. Instead of plotting lots of coordinates, just follow where (0,0) ends up - this tells you everything about the shift.
Pro Strategy: Always identify the minimum point first - it's like having a GPS tracker for your parabola!
Move Left or Right
Horizontal shifts might seem backwards at first, but there's a simple rule to remember. When you see ², the graph moves left by 'a' units. When you see ², it moves right by 'a' units.
For example, ² shifts the parabola 3 units to the right, placing the minimum point at (3,0). The sign inside the brackets works opposite to the direction you might expect.
Think of it as the x-value that makes the bracket equal zero - that's where your new minimum point sits horizontally.
Memory Trick: The sign is opposite to the movement - plus means left, minus means right!
Move Up or Down
Vertical shifts are much more straightforward than horizontal ones. Adding a number outside the squared term moves the graph up, whilst subtracting moves it down.
The form (x)² + a shifts up by 'a' units, and (x)² - a shifts down by 'a' units. For instance, (x)² - 2 drops the entire parabola down 2 units.
These vertical transformations change the y-coordinate of every point on the curve by the same amount.
Easy Rule: What you see is what you get - plus goes up, minus goes down!
Stretch and Compression in the x Direction
Horizontal stretching and compression happen when you multiply x by a factor inside the function. The form f(ax) creates compression when a > 1, and f creates stretching.
For example, y = (2x)² squashes the parabola horizontally, making it narrower. Meanwhile, y = ² stretches it out, making it wider.
This can feel counterintuitive because larger numbers inside actually make the graph narrower, whilst fractions make it wider.
Key Insight: Think about what x-values give the same y-output - that reveals the stretch factor!
Stretch and Compression in the y Direction
Vertical transformations multiply the entire function by a factor. The form af(x) stretches vertically when a > 1, and compresses when 0 < a < 1.
For example, y = 5x² makes the parabola five times taller, whilst y = (1/5)x² squashes it down to one-fifth of its original height.
These changes affect every y-coordinate on the graph, multiplying each one by your transformation factor.
Visual Check: Vertical stretches make graphs look skinnier, compressions make them look flatter!
Reflection
Reflections flip your graph across an axis. The form f reflects across the y-axis, whilst -f(x) reflects across the x-axis.
For parabolas like y = x², reflecting in the y-axis doesn't change anything because they're symmetrical. However, y = -(x)² flips it upside down, creating an upside-down parabola.
These transformations are like holding up a mirror to your graph - every point flips to the opposite side of the axis.
Mirror Rule: Negative outside flips vertically, negative inside flips horizontally!
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Stefan S
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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
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Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Mastering Curve Transformations with Completing the Square
Ever wondered how graphs move and change shape on a coordinate plane? Transforming curves is all about shifting, stretching, and flipping graphs - and once you get the hang of it, you'll be able to predict exactly where any curve... Show more
Transforming Curves
Understanding how curves transform is like learning the secret language of graphs. You'll see how equations change to create different positions and shapes on your coordinate system.
The key to mastering transformations is recognising patterns in equations and linking them to visual changes on the graph. Once you spot these patterns, you can predict transformations without even drawing the curve!
Quick Tip: Think of transformations as giving directions to a graph - "move left 3 units" or "stretch by factor 2" - the maths just codes these instructions!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Complete the Square
Completing the square is your essential first step for spotting transformations. This technique converts messy quadratic expressions into a neat form that reveals exactly how the graph has moved.
Here's the process: take half the coefficient of x, square it, then subtract this value whilst adding the constant. For example, with x² + 8x - 3, you get ² - 19.
The result always follows the pattern (x ± a)² ± b, which immediately shows you the horizontal and vertical shifts of your parabola.
Remember: Half the coefficient, square it, subtract it - this sequence becomes automatic with practice!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Before a Transformation
The basic graph y = x² has its minimum point at the origin (0,0). This is your starting reference point for all transformations.
Tracking how this minimum point moves is the smartest way to understand transformations. Instead of plotting lots of coordinates, just follow where (0,0) ends up - this tells you everything about the shift.
Pro Strategy: Always identify the minimum point first - it's like having a GPS tracker for your parabola!
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Move Left or Right
Horizontal shifts might seem backwards at first, but there's a simple rule to remember. When you see ², the graph moves left by 'a' units. When you see ², it moves right by 'a' units.
For example, ² shifts the parabola 3 units to the right, placing the minimum point at (3,0). The sign inside the brackets works opposite to the direction you might expect.
Think of it as the x-value that makes the bracket equal zero - that's where your new minimum point sits horizontally.
Memory Trick: The sign is opposite to the movement - plus means left, minus means right!
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Move Up or Down
Vertical shifts are much more straightforward than horizontal ones. Adding a number outside the squared term moves the graph up, whilst subtracting moves it down.
The form (x)² + a shifts up by 'a' units, and (x)² - a shifts down by 'a' units. For instance, (x)² - 2 drops the entire parabola down 2 units.
These vertical transformations change the y-coordinate of every point on the curve by the same amount.
Easy Rule: What you see is what you get - plus goes up, minus goes down!
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Stretch and Compression in the x Direction
Horizontal stretching and compression happen when you multiply x by a factor inside the function. The form f(ax) creates compression when a > 1, and f creates stretching.
For example, y = (2x)² squashes the parabola horizontally, making it narrower. Meanwhile, y = ² stretches it out, making it wider.
This can feel counterintuitive because larger numbers inside actually make the graph narrower, whilst fractions make it wider.
Key Insight: Think about what x-values give the same y-output - that reveals the stretch factor!
Sign up to see the contentIt's free!
Access to all documents
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Stretch and Compression in the y Direction
Vertical transformations multiply the entire function by a factor. The form af(x) stretches vertically when a > 1, and compresses when 0 < a < 1.
For example, y = 5x² makes the parabola five times taller, whilst y = (1/5)x² squashes it down to one-fifth of its original height.
These changes affect every y-coordinate on the graph, multiplying each one by your transformation factor.
Visual Check: Vertical stretches make graphs look skinnier, compressions make them look flatter!
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Reflection
Reflections flip your graph across an axis. The form f reflects across the y-axis, whilst -f(x) reflects across the x-axis.
For parabolas like y = x², reflecting in the y-axis doesn't change anything because they're symmetrical. However, y = -(x)² flips it upside down, creating an upside-down parabola.
These transformations are like holding up a mirror to your graph - every point flips to the opposite side of the axis.
Mirror Rule: Negative outside flips vertically, negative inside flips horizontally!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user