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2m wars and welfare: britain in transition, 1906-1957
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Kate
15/09/2023
Maths
AQA Pure Math- Algebra and Functions- Year 1 and 2
This comprehensive guide covers key topics in AQA Year 1 and 2 Pure Math Algebra and Functions, providing essential notes and examples for students preparing for their A Level Maths exams. The document covers a wide range of algebraic concepts and techniques, from basic laws of indices to complex function transformations.
15/09/2023
511
This section delves deeper into quadratic equations, introducing the discriminant and its role in determining the nature of roots. It also covers completing the square and solving hidden quadratics. The page concludes with an introduction to simultaneous equations.
Vocabulary: The discriminant is the expression b² - 4ac in a quadratic equation ax² + bx + c = 0.
Example: For the equation 3x² + 1 - 8 × 3^x + 27 = 0, we can rewrite it as a quadratic in terms of 3^x: 3² - 8 + 27 = 0.
Highlight: The discriminant helps determine the nature of roots: b² - 4ac > 0 indicates two real roots, b² - 4ac = 0 suggests one real root, and b² - 4ac < 0 means no real roots.
The page also introduces the quadratic formula and explains how to use it to solve quadratic equations. This is a crucial tool for A Level Maths students, often appearing in exam questions.
This page covers inequalities, including quadratic inequalities, and introduces polynomials and algebraic division. It explains how to solve and represent inequalities graphically, which is a common topic in A Level Maths questions and answers.
Definition: A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.
Example: To solve the quadratic inequality 2 > 0, first solve the equation 2 = 0, then plot the results on a number line to determine the solution.
Highlight: When representing inequalities on a graph, use solid lines for ≤ or ≥, and dotted lines for < or >.
The page also covers algebraic division, an essential technique for factoring higher-degree polynomials and solving more complex equations.
This section focuses on the Factor Theorem and its application in factorising polynomials. It provides a step-by-step guide to using the Factor Theorem in combination with algebraic division to fully factorise polynomial expressions.
Definition: The Factor Theorem states that is a factor of f if and only if f = 0.
Example: To factorise x³ + 4x² - 11x - 10, first find a factor using the Factor Theorem, then use algebraic division to find the remaining quadratic factor.
Highlight: The Factor Theorem is a powerful tool for factorising higher-degree polynomials, which is crucial for solving complex equations in A Level Maths.
The page also covers algebraic fractions, explaining how to simplify them by factorising both the numerator and denominator and cancelling common factors.
This page introduces improper algebraic fractions and their division process. It also covers graphing functions, including finding turning points, y-intercepts, and x-intercepts .
Vocabulary: An improper algebraic fraction is one where the degree of the numerator is greater than or equal to the degree of the denominator.
Example: To divide the improper fraction ÷ , perform polynomial long division to get a quotient and remainder.
Highlight: When graphing functions, remember that y-intercepts are found by setting x = 0, while x-intercepts are found by setting y = 0.
The page provides a comprehensive guide to analysing function graphs, which is essential for understanding the behavior of various functions in A Level Maths.
This section covers reciprocal graphs and solving equations graphically. It explains the characteristics of reciprocal graphs and provides examples of how to solve equations by finding the intersection points of graphs.
Definition: A reciprocal graph is the graph of a function in the form y = k/x, where k is a constant.
Example: To solve the equation 2x³ + 5x² + 2x = 0 graphically, plot y = 2x³ + 5x² + 2x and y = 0, and find their points of intersection.
Highlight: Reciprocal graphs always have asymptotes at x = 0 and y = 0, and their shape depends on whether k is positive or negative.
The page also covers solving more complex equations graphically by finding the intersection points of two or more graphs, a technique often used in A Level Maths exam questions.
This page introduces proportional relationships and functions, including direct proportion, inverse proportion, and composite functions. It also covers function notation, domain and range, and mapping diagrams.
Vocabulary: The domain of a function is the set of all possible input values, while the range is the set of all possible output values.
Example: For the function f = x², with domain -5 ≤ x ≤ 5, the range would be 0 ≤ y ≤ 25.
Highlight: Composite functions involve applying one function after another, often written as f).
The page also introduces the concepts of one-to-one and many-to-one functions, which are important for understanding function behavior and invertibility.
This section focuses on modulus functions and their graphs. It explains how to sketch modulus function graphs and solve modulus equations. The page also covers function transformations, including translations and stretches.
Definition: The modulus function |x| returns the non-negative value of x without regard to its sign.
Example: To sketch y = |f|, first sketch y = f, then reflect any part of the graph below the x-axis in the x-axis.
Highlight: When solving modulus equations like |2x - 4| = |x - 1|, consider both positive and negative cases for each modulus expression.
The page provides detailed explanations of various function transformations, which are crucial for understanding how changes to function equations affect their graphs.
The final page of the document continues the discussion on function transformations, focusing on translations and stretches. It provides examples of how these transformations affect function graphs.
Vocabulary: A translation moves every point of a graph by the same distance in a given direction, while a stretch enlarges or shrinks the graph by a certain factor.
Example: The graph of y = f + 1 is a vertical translation of y = f by 1 unit upwards.
Highlight: Horizontal translations work in the opposite direction to what you might expect: y = f shifts the graph a units to the left.
The page concludes with a summary of how different transformations affect function graphs, providing a comprehensive overview of this important topic in A Level Maths.
This page introduces fundamental algebraic concepts and techniques crucial for AQA A Level Maths students. It covers laws of indices, manipulation of surds, and quadratic graphs. The page also explains how to find roots, turning points, and y-intercepts of quadratic equations.
Definition: Laws of indices are rules for simplifying expressions involving powers, such as a^m × a^n = a^.
Example: To find the roots of a quadratic equation y = ax² + bx + c, you can either factorise or use the quadratic formula.
Highlight: The shape of a quadratic graph depends on the sign of 'a' in the equation y = ax² + bx + c. Positive 'a' results in a U-shaped graph, while negative 'a' produces an inverted U-shape.
The page also introduces the concept of rationalising the denominator when dealing with surds, which is an important technique in simplifying algebraic expressions.
1655
20073
12
Edexcel Year 1 Maths - Complete notes!
ignore the rough start but these notes have helped me get an A* Prediction
5
694
12/13
Simultaneous Equations Questions
Edexcel A Level Maths simultaneous equations questions
10
892
S5/S6
SQA HIGHER Mathematics 2019 P2 Q11 (OPTIMISATION question)
Worked solutions to part A and B of the 2019 higher maths paper 2 differentiation optimisation question. Part A can be tricky at first and there is a lot of work to do for just 3 marks but I have tried my best to break it down and explain it.
8
189
12/13
Quadratics and Modelling Questions
Edexcel A-Level Maths quadratics and modelling questions
10
139
12
As pure edexcel proof
3 types of proof with examples
2
147
12/13
Polynomial division
Notes, explanations and practice questions (with steps) on polynomial division
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Kate
@katerevisionotes
This comprehensive guide covers key topics in AQA Year 1 and 2 Pure Math Algebra and Functions, providing essential notes and examples for students preparing for their A Level Mathsexams. The document covers a wide range of algebraic... Show more
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This section delves deeper into quadratic equations, introducing the discriminant and its role in determining the nature of roots. It also covers completing the square and solving hidden quadratics. The page concludes with an introduction to simultaneous equations.
Vocabulary: The discriminant is the expression b² - 4ac in a quadratic equation ax² + bx + c = 0.
Example: For the equation 3x² + 1 - 8 × 3^x + 27 = 0, we can rewrite it as a quadratic in terms of 3^x: 3² - 8 + 27 = 0.
Highlight: The discriminant helps determine the nature of roots: b² - 4ac > 0 indicates two real roots, b² - 4ac = 0 suggests one real root, and b² - 4ac < 0 means no real roots.
The page also introduces the quadratic formula and explains how to use it to solve quadratic equations. This is a crucial tool for A Level Maths students, often appearing in exam questions.
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Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page covers inequalities, including quadratic inequalities, and introduces polynomials and algebraic division. It explains how to solve and represent inequalities graphically, which is a common topic in A Level Maths questions and answers.
Definition: A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.
Example: To solve the quadratic inequality 2 > 0, first solve the equation 2 = 0, then plot the results on a number line to determine the solution.
Highlight: When representing inequalities on a graph, use solid lines for ≤ or ≥, and dotted lines for < or >.
The page also covers algebraic division, an essential technique for factoring higher-degree polynomials and solving more complex equations.
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This section focuses on the Factor Theorem and its application in factorising polynomials. It provides a step-by-step guide to using the Factor Theorem in combination with algebraic division to fully factorise polynomial expressions.
Definition: The Factor Theorem states that is a factor of f if and only if f = 0.
Example: To factorise x³ + 4x² - 11x - 10, first find a factor using the Factor Theorem, then use algebraic division to find the remaining quadratic factor.
Highlight: The Factor Theorem is a powerful tool for factorising higher-degree polynomials, which is crucial for solving complex equations in A Level Maths.
The page also covers algebraic fractions, explaining how to simplify them by factorising both the numerator and denominator and cancelling common factors.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page introduces improper algebraic fractions and their division process. It also covers graphing functions, including finding turning points, y-intercepts, and x-intercepts .
Vocabulary: An improper algebraic fraction is one where the degree of the numerator is greater than or equal to the degree of the denominator.
Example: To divide the improper fraction ÷ , perform polynomial long division to get a quotient and remainder.
Highlight: When graphing functions, remember that y-intercepts are found by setting x = 0, while x-intercepts are found by setting y = 0.
The page provides a comprehensive guide to analysing function graphs, which is essential for understanding the behavior of various functions in A Level Maths.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This section covers reciprocal graphs and solving equations graphically. It explains the characteristics of reciprocal graphs and provides examples of how to solve equations by finding the intersection points of graphs.
Definition: A reciprocal graph is the graph of a function in the form y = k/x, where k is a constant.
Example: To solve the equation 2x³ + 5x² + 2x = 0 graphically, plot y = 2x³ + 5x² + 2x and y = 0, and find their points of intersection.
Highlight: Reciprocal graphs always have asymptotes at x = 0 and y = 0, and their shape depends on whether k is positive or negative.
The page also covers solving more complex equations graphically by finding the intersection points of two or more graphs, a technique often used in A Level Maths exam questions.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page introduces proportional relationships and functions, including direct proportion, inverse proportion, and composite functions. It also covers function notation, domain and range, and mapping diagrams.
Vocabulary: The domain of a function is the set of all possible input values, while the range is the set of all possible output values.
Example: For the function f = x², with domain -5 ≤ x ≤ 5, the range would be 0 ≤ y ≤ 25.
Highlight: Composite functions involve applying one function after another, often written as f).
The page also introduces the concepts of one-to-one and many-to-one functions, which are important for understanding function behavior and invertibility.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This section focuses on modulus functions and their graphs. It explains how to sketch modulus function graphs and solve modulus equations. The page also covers function transformations, including translations and stretches.
Definition: The modulus function |x| returns the non-negative value of x without regard to its sign.
Example: To sketch y = |f|, first sketch y = f, then reflect any part of the graph below the x-axis in the x-axis.
Highlight: When solving modulus equations like |2x - 4| = |x - 1|, consider both positive and negative cases for each modulus expression.
The page provides detailed explanations of various function transformations, which are crucial for understanding how changes to function equations affect their graphs.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The final page of the document continues the discussion on function transformations, focusing on translations and stretches. It provides examples of how these transformations affect function graphs.
Vocabulary: A translation moves every point of a graph by the same distance in a given direction, while a stretch enlarges or shrinks the graph by a certain factor.
Example: The graph of y = f + 1 is a vertical translation of y = f by 1 unit upwards.
Highlight: Horizontal translations work in the opposite direction to what you might expect: y = f shifts the graph a units to the left.
The page concludes with a summary of how different transformations affect function graphs, providing a comprehensive overview of this important topic in A Level Maths.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page introduces fundamental algebraic concepts and techniques crucial for AQA A Level Maths students. It covers laws of indices, manipulation of surds, and quadratic graphs. The page also explains how to find roots, turning points, and y-intercepts of quadratic equations.
Definition: Laws of indices are rules for simplifying expressions involving powers, such as a^m × a^n = a^.
Example: To find the roots of a quadratic equation y = ax² + bx + c, you can either factorise or use the quadratic formula.
Highlight: The shape of a quadratic graph depends on the sign of 'a' in the equation y = ax² + bx + c. Positive 'a' results in a U-shaped graph, while negative 'a' produces an inverted U-shape.
The page also introduces the concept of rationalising the denominator when dealing with surds, which is an important technique in simplifying algebraic expressions.
Access to all documents
Improve your grades
Join milions of students
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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
