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Math (SAT®)
Dec 3, 2025
103
4 pages
These study notes cover essential algebra concepts that form the foundation of high school math. From linear equations... Show more
Linear equations are the backbone of algebra and appear in several different formats. The most common form is slope-intercept form , where m represents the slope and b is the y-intercept.
You can also write linear equations in standard form or point-slope form . The slope of a line is calculated as the ratio of rise over run or /.
To create an equation from two points, first find the slope, then substitute into slope-intercept form. Next, use one of your points to solve for b, and finally write your complete equation. Remember that horizontal lines have a slope of 0, while vertical lines have an undefined slope.
💡 When converting from standard form to slope-intercept form, solve for y to get y = -A/B·x + C/B. This immediately shows you the slope and y-intercept .
Functions assign exactly one y-value to each x-value. We write f(3) to represent the y-value when x equals 3, and f(g(x)) means we plug the entire g(x) expression in as the x-value for function f.
To determine if a point (x,y) lies on a line, simply substitute the x and y values into the equation. If the equation becomes a true statement, the point is on the line. For example, checking if (3,4) is on y = 2x 4 = 2(3) becomes 4 = 6, which is false.
Perpendicular lines have slopes that are negative reciprocals of each other. For instance, if one line has a slope of 2, any perpendicular line will have a slope of -1/2.
🔍 Synthetic division is a shortcut method for dividing polynomials. It's especially useful when applying the remainder theorem—if you divide a polynomial by , the remainder equals the value of the polynomial when x = a.
Quadratic equations (containing x²) can be solved by factoring or using the quadratic formula. The discriminant tells you the nature of the solutions positive discriminant means two real solutions, zero means one real solution, and negative means complex solutions.
The vertex of a parabola represents its minimum or maximum point. Find the x-coordinate using x = -b/2a or the average of the two roots /2. Then plug this value back into the original equation to find the y-coordinate.
Quadratics can be written in vertex form y = a² + k, where (h,k) is the vertex, or in factored form which clearly shows the x-intercepts. You can also find the sum of roots using -b/a and the product of roots using c/a.
🧩 The vertex form is perfect for finding minimum/maximum values, while the factored form makes it easy to identify where the parabola crosses the x-axis. Choose the form that best suits what you're trying to find!
Complex numbers involve the imaginary unit i, where i² = -1. When working with powers of i, you'll notice a pattern i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats every four powers.
Circle equations follow the form ² + ² = r², where (h,k) is the center point and r is the radius. This is derived from the distance formula between any point on the circle and its center.
For parts of circles, calculate arc length using 2πr · (θ/360) and sector area using πr² · (θ/360), where θ is the central angle in degrees. These formulas help you work with portions of circles in various applications.
🔄 Complex numbers may seem strange at first, but they become more intuitive with practice. Just remember to treat i like a variable, and follow the pattern of powers as you work through calculations.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
7
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
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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
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
These study notes cover essential algebra concepts that form the foundation of high school math. From linear equations and functions to quadratics and complex numbers, mastering these topics will help you solve mathematical problems across various contexts.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Linear equations are the backbone of algebra and appear in several different formats. The most common form is slope-intercept form , where m represents the slope and b is the y-intercept.
You can also write linear equations in standard form or point-slope form . The slope of a line is calculated as the ratio of rise over run or /.
To create an equation from two points, first find the slope, then substitute into slope-intercept form. Next, use one of your points to solve for b, and finally write your complete equation. Remember that horizontal lines have a slope of 0, while vertical lines have an undefined slope.
💡 When converting from standard form to slope-intercept form, solve for y to get: y = -A/B·x + C/B. This immediately shows you the slope and y-intercept .
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Functions assign exactly one y-value to each x-value. We write f(3) to represent the y-value when x equals 3, and f(g(x)) means we plug the entire g(x) expression in as the x-value for function f.
To determine if a point (x,y) lies on a line, simply substitute the x and y values into the equation. If the equation becomes a true statement, the point is on the line. For example, checking if (3,4) is on y = 2x: 4 = 2(3) becomes 4 = 6, which is false.
Perpendicular lines have slopes that are negative reciprocals of each other. For instance, if one line has a slope of 2, any perpendicular line will have a slope of -1/2.
🔍 Synthetic division is a shortcut method for dividing polynomials. It's especially useful when applying the remainder theorem—if you divide a polynomial by , the remainder equals the value of the polynomial when x = a.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Quadratic equations (containing x²) can be solved by factoring or using the quadratic formula. The discriminant tells you the nature of the solutions: positive discriminant means two real solutions, zero means one real solution, and negative means complex solutions.
The vertex of a parabola represents its minimum or maximum point. Find the x-coordinate using x = -b/2a or the average of the two roots /2. Then plug this value back into the original equation to find the y-coordinate.
Quadratics can be written in vertex form y = a² + k, where (h,k) is the vertex, or in factored form which clearly shows the x-intercepts. You can also find the sum of roots using -b/a and the product of roots using c/a.
🧩 The vertex form is perfect for finding minimum/maximum values, while the factored form makes it easy to identify where the parabola crosses the x-axis. Choose the form that best suits what you're trying to find!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Complex numbers involve the imaginary unit i, where i² = -1. When working with powers of i, you'll notice a pattern: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats every four powers.
Circle equations follow the form ² + ² = r², where (h,k) is the center point and r is the radius. This is derived from the distance formula between any point on the circle and its center.
For parts of circles, calculate arc length using 2πr · (θ/360) and sector area using πr² · (θ/360), where θ is the central angle in degrees. These formulas help you work with portions of circles in various applications.
🔄 Complex numbers may seem strange at first, but they become more intuitive with practice. Just remember to treat i like a variable, and follow the pattern of powers as you work through calculations.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
7
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
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
Algebra 1
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