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9
1
B
Bethyn King
08/09/2022
Maths
CCEA MATHS AS NOTES
254
•
8 Sept 2022
•
Fun CCEA Maths: Curve Sketching and Quadratics for Kids
B
Bethyn King
@bethyn_k.04
1 / 3
Completing the Square and Solving Quadratics
This page focuses on the technique of completing the square, a crucial method in CCEA GCE Maths for solving quadratic equations and finding minimum or maximum values of quadratic functions.
The process of completing the square is explained step-by-step:
- Rewrite the quadratic in the form a² + c
- Use this form to find the minimum or maximum value
- Solve the equation by setting it equal to zero
Example: For 2x² - 8x + 7, completing the square gives 2² - 1. The minimum value is -1 when x = 2.
The page also covers solving quadratic equations using the completed square form, leaving answers in surd form when necessary.
Vocabulary: Surd form refers to expressions involving square roots that cannot be simplified further.
Additional topics covered include:
- Indices and their properties
- Solving quadratics by factoring
- Simplifying and manipulating surds
Highlight: Understanding how to complete the square is essential for finding the vertex of a parabola and solving quadratic equations that cannot be easily factored.
The page provides numerous examples and practice problems to reinforce these concepts, making it an excellent resource for CCEA A Level Maths students preparing for exams or seeking to deepen their understanding of quadratic functions.
Advanced Quadratic Techniques and Curve Sketching
This final page delves deeper into completing the square and its applications in curve sketching for CCEA GCE Maths. It demonstrates how to use this technique to solve more complex quadratic equations and find maximum or minimum values of quadratic expressions.
Example: For x² + 7x - 11 = 0, completing the square yields ² = 93/4, leading to the solution x = -7/2 ± √(93/4).
The page emphasizes the utility of completing the square in finding maximum or minimum values of quadratic expressions:
Highlight: Completing the square transforms a quadratic expression into a form that makes it easy to identify its extreme value and where it occurs.
Several examples are provided, such as:
- Finding the minimum value of x² - 6x + 6
- Determining the maximum value of 8 - 2x - x²
The final section focuses on sketching quadratic functions, integrating all the techniques learned:
- Finding roots of the function
- Identifying the maximum or minimum point
- Determining the y-intercept
- Sketching the parabola
Example: For y = x² - 7x - 4, the roots are approximately 7.55 and -0.55, the minimum point is at , and the y-intercept is at .
This comprehensive approach to curve sketching ties together multiple concepts from the CCEA A Level Maths specification, providing students with a robust method for analyzing and graphing quadratic functions.
Vocabulary: Parabola - the U-shaped curve that represents a quadratic function graphically.
The page concludes with practice problems, reinforcing the importance of these techniques in the CCEA GCE Maths curriculum and preparing students for potential CCEA GCE Maths SPECIMEN PAPER questions.
Curve Sketching and Quadratic Equations
This page covers advanced techniques for curve sketching and solving quadratic equations in CCEA GCE Maths. It begins with an example of sketching the curve y = -x² - 6x + 7, demonstrating how to find roots and the maximum point.
Example: For y = -x² - 6x + 7, the roots are found at x = -7 and x = 1, and the maximum point is at .
The page then introduces the discriminant and its role in determining the nature of roots for quadratic equations.
Definition: The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac.
Different cases for the discriminant are explained:
- b² - 4ac > 0: Two real roots
- b² - 4ac = 0: One repeated root
- b² - 4ac < 0: No real roots
Examples are provided to illustrate how to use the discriminant to determine the number of distinct roots for given quadratic equations.
Highlight: The discriminant is a powerful tool for analyzing the nature of roots in quadratic equations without solving them explicitly.
The page concludes with an exercise on finding values of k for which a quadratic equation has equal roots, demonstrating the practical application of the discriminant concept.
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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
Fun CCEA Maths: Curve Sketching and Quadratics for Kids
B
Bethyn King
@bethyn_k.04
The document provides comprehensive CCEA GCE Maths notes on curve sketching and solving quadratic equations. It covers essential topics like completing the square, using the discriminant, and working with surds. The material is suitable for students preparing for CCEA A... Show more
Completing the Square and Solving Quadratics
This page focuses on the technique of completing the square, a crucial method in CCEA GCE Maths for solving quadratic equations and finding minimum or maximum values of quadratic functions.
The process of completing the square is explained step-by-step:
- Rewrite the quadratic in the form a² + c
- Use this form to find the minimum or maximum value
- Solve the equation by setting it equal to zero
Example: For 2x² - 8x + 7, completing the square gives 2² - 1. The minimum value is -1 when x = 2.
The page also covers solving quadratic equations using the completed square form, leaving answers in surd form when necessary.
Vocabulary: Surd form refers to expressions involving square roots that cannot be simplified further.
Additional topics covered include:
- Indices and their properties
- Solving quadratics by factoring
- Simplifying and manipulating surds
Highlight: Understanding how to complete the square is essential for finding the vertex of a parabola and solving quadratic equations that cannot be easily factored.
The page provides numerous examples and practice problems to reinforce these concepts, making it an excellent resource for CCEA A Level Maths students preparing for exams or seeking to deepen their understanding of quadratic functions.
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
Advanced Quadratic Techniques and Curve Sketching
This final page delves deeper into completing the square and its applications in curve sketching for CCEA GCE Maths. It demonstrates how to use this technique to solve more complex quadratic equations and find maximum or minimum values of quadratic expressions.
Example: For x² + 7x - 11 = 0, completing the square yields ² = 93/4, leading to the solution x = -7/2 ± √(93/4).
The page emphasizes the utility of completing the square in finding maximum or minimum values of quadratic expressions:
Highlight: Completing the square transforms a quadratic expression into a form that makes it easy to identify its extreme value and where it occurs.
Several examples are provided, such as:
- Finding the minimum value of x² - 6x + 6
- Determining the maximum value of 8 - 2x - x²
The final section focuses on sketching quadratic functions, integrating all the techniques learned:
- Finding roots of the function
- Identifying the maximum or minimum point
- Determining the y-intercept
- Sketching the parabola
Example: For y = x² - 7x - 4, the roots are approximately 7.55 and -0.55, the minimum point is at , and the y-intercept is at .
This comprehensive approach to curve sketching ties together multiple concepts from the CCEA A Level Maths specification, providing students with a robust method for analyzing and graphing quadratic functions.
Vocabulary: Parabola - the U-shaped curve that represents a quadratic function graphically.
The page concludes with practice problems, reinforcing the importance of these techniques in the CCEA GCE Maths curriculum and preparing students for potential CCEA GCE Maths SPECIMEN PAPER questions.
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
Curve Sketching and Quadratic Equations
This page covers advanced techniques for curve sketching and solving quadratic equations in CCEA GCE Maths. It begins with an example of sketching the curve y = -x² - 6x + 7, demonstrating how to find roots and the maximum point.
Example: For y = -x² - 6x + 7, the roots are found at x = -7 and x = 1, and the maximum point is at .
The page then introduces the discriminant and its role in determining the nature of roots for quadratic equations.
Definition: The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac.
Different cases for the discriminant are explained:
- b² - 4ac > 0: Two real roots
- b² - 4ac = 0: One repeated root
- b² - 4ac < 0: No real roots
Examples are provided to illustrate how to use the discriminant to determine the number of distinct roots for given quadratic equations.
Highlight: The discriminant is a powerful tool for analyzing the nature of roots in quadratic equations without solving them explicitly.
The page concludes with an exercise on finding values of k for which a quadratic equation has equal roots, demonstrating the practical application of the discriminant concept.
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?
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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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