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189

27 Nov 2025

4 pages

Higher Math Quick Review Cards

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Evie

@eviechristie_qmje

This study guide covers essential A-level maths topics you'll definitely... Show more

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1 / 4
QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

Sketching Quadratic Parabolas

Ever wondered how to tackle those "sketch the graph" questions without panicking? Quadratic functions like y = x² + 6x + 8 follow a simple pattern once you know what to look for.

Start by identifying whether your parabola opens upward UshapedU-shaped or downward shaped∩-shaped - just check if the coefficient of x² is positive or negative. Next, find your y-intercept by substituting x = 0 into the equation.

The roots are where your graph crosses the x-axis wheny=0when y = 0. Factor the quadratic to find these x-values quickly. Finally, the axis of symmetry sits exactly halfway between your roots, and that's where you'll find your turning point.

Quick tip: The turning point's x-coordinate is always the average of your two roots!

QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

Completing the Square

Completing the square transforms messy quadratics into neat, workable forms that reveal everything about your parabola instantly. You're converting y = ax² + bx + c into the much more useful form y = a(x ± p)² + q.

Here's the magic formula: take half of your b coefficient, square it, then add and subtract this value cleverly. For x² + 8x + 3, you'd use (8/2)² = 16 to create the perfect square x+4x + 4².

This method becomes incredibly powerful when finding roots of complex equations. Once you've completed the square, you can solve by taking square roots rather than struggling with factoring.

Remember: When the coefficient of x² isn't 1, factor it out first before completing the square!

QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

Exact Trigonometric Values

Those special angles - 30°, 45°, and 60° - pop up everywhere in A-level maths, so memorising their exact values is absolutely crucial for your success.

The 45-45-90 triangle gives you equal sides of 1 and a hypotenuse of √2. This means sin 45° = cos 45° = 1/√2 or2/2whenrationalisedor √2/2 when rationalised.

The 30-60-90 triangle has sides in the ratio 1:√3:2. From this, sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, and cos 60° = 1/2. These values are worth their weight in gold during exams!

Pro tip: Draw these triangles in your exam booklet if you forget - the side lengths will remind you of all the ratios instantly!

QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

Distance Formula

The distance formula is your go-to tool for finding the length between any two points on a coordinate plane - think of it as Pythagoras' theorem in disguise.

Simply plug your coordinates (x₁, y₁) and (x₂, y₂) into √(x2x1)2+(y2y1)2(x₂ - x₁)² + (y₂ - y₁)². The formula calculates the hypotenuse of the right triangle formed by the horizontal and vertical distances between your points.

Quick check: Always ensure your final answer makes sense by roughly estimating the distance on a sketch!



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.

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

 

Maths

189

27 Nov 2025

4 pages

Higher Math Quick Review Cards

user profile picture

Evie

@eviechristie_qmje

This study guide covers essential A-level maths topics you'll definitely encounter in your exams. From sketching parabolas to finding exact trigonometric values, these are the fundamental skills that'll boost your confidence in tackling those tricky questions.

QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

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

Sketching Quadratic Parabolas

Ever wondered how to tackle those "sketch the graph" questions without panicking? Quadratic functions like y = x² + 6x + 8 follow a simple pattern once you know what to look for.

Start by identifying whether your parabola opens upward UshapedU-shaped or downward shaped∩-shaped - just check if the coefficient of x² is positive or negative. Next, find your y-intercept by substituting x = 0 into the equation.

The roots are where your graph crosses the x-axis wheny=0when y = 0. Factor the quadratic to find these x-values quickly. Finally, the axis of symmetry sits exactly halfway between your roots, and that's where you'll find your turning point.

Quick tip: The turning point's x-coordinate is always the average of your two roots!

QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

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

Completing the Square

Completing the square transforms messy quadratics into neat, workable forms that reveal everything about your parabola instantly. You're converting y = ax² + bx + c into the much more useful form y = a(x ± p)² + q.

Here's the magic formula: take half of your b coefficient, square it, then add and subtract this value cleverly. For x² + 8x + 3, you'd use (8/2)² = 16 to create the perfect square x+4x + 4².

This method becomes incredibly powerful when finding roots of complex equations. Once you've completed the square, you can solve by taking square roots rather than struggling with factoring.

Remember: When the coefficient of x² isn't 1, factor it out first before completing the square!

QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

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

Exact Trigonometric Values

Those special angles - 30°, 45°, and 60° - pop up everywhere in A-level maths, so memorising their exact values is absolutely crucial for your success.

The 45-45-90 triangle gives you equal sides of 1 and a hypotenuse of √2. This means sin 45° = cos 45° = 1/√2 or2/2whenrationalisedor √2/2 when rationalised.

The 30-60-90 triangle has sides in the ratio 1:√3:2. From this, sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, and cos 60° = 1/2. These values are worth their weight in gold during exams!

Pro tip: Draw these triangles in your exam booklet if you forget - the side lengths will remind you of all the ratios instantly!

QUADRATICS
|
Ex: Sketch the graph
y = x ² + 6x + 8
PARABOLAS
Uo A
or
|_=+x² = -x²
1
|
1y = intercept ! 50=0
y = 0² + 6(0) + 8
1
y=8
(0,8)
Sk

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

Distance Formula

The distance formula is your go-to tool for finding the length between any two points on a coordinate plane - think of it as Pythagoras' theorem in disguise.

Simply plug your coordinates (x₁, y₁) and (x₂, y₂) into √(x2x1)2+(y2y1)2(x₂ - x₁)² + (y₂ - y₁)². The formula calculates the hypotenuse of the right triangle formed by the horizontal and vertical distances between your points.

Quick check: Always ensure your final answer makes sense by roughly estimating the distance on a sketch!

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.

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