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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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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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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!
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
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
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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 a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
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Most popular content: Completing the Square
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Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user