Biology
Chemistry
AP Chemistry
English
AP English Literature
Algebra 1
Dec 16, 2025
98
3 pages
Catalina Collazos @catalinacollazos9
Quadratic functions are equations in the form of f(x) = ax² + bx + c that create parabola-shaped... Show more
A quadratic function follows the form f(x) = ax² + bx + c, where a, b, and c are constants. Each quadratic creates a parabola when graphed, with several important features to identify.
The vertex is the highest or lowest point on the parabola, found at . For example, in f(x) = 2x² - 4x + 3, the vertex sits at (1,1). This point tells you where the function reaches its maximum or minimum value.
The axis of symmetry is a vertical line passing through the vertex, dividing the parabola into mirror images. It's always at x = -b/2a. In f(x) = -x² + 6x - 5, the axis of symmetry is x = 3.
💡 Think of a quadratic function like a mirror - the axis of symmetry is where you'd place the mirror to create identical reflections on both sides!
The discriminant tells you about the roots of the equation. For f(x) = 3x² + 2x + 1, the discriminant is D = -8, which means this quadratic has no real roots .
The vertex form of a quadratic function is f(x) = a² + k, where (h,k) is the vertex. This form makes identifying the vertex simple! For example, f(x) = 4x² + 8x + 5 can be rewritten as f(x) = 4² + 1, showing the vertex is at (-1,1).
The direction of the parabola depends on the value of a. When a > 0, the parabola opens upward (like a cup). When a < 0, it opens downward (like an inverted cup). For instance, f(x) = -x² + 2x + 1 opens downward because a = -1.
The y-intercept occurs where the parabola crosses the y-axis, always at (0,c). For f(x) = 2x² + 3x - 1, the y-intercept is (0,-1).
🔍 The sign of the leading coefficient (a) tells you everything about the parabola's direction - it's like gravity either pulling down (a > 0) or pushing up (a < 0)!
The roots or zeros are where f(x) = 0, found using the quadratic formula x = /2a. For example, f(x) = x² - 4x + 3 has roots at x = 1 and x = 3.
Every quadratic function has either a maximum or minimum value at its vertex. When the parabola opens downward (a < 0), the vertex is a maximum. When it opens upward (a > 0), the vertex is a minimum. For f(x) = -2x² + 4x + 7, the maximum value is 9, occurring at x = 1.
The shape of the parabola depends primarily on the value of a. A larger |a| makes the parabola narrower, while a smaller |a| makes it wider. Compare f(x) = x² (standard upward parabola) with f(x) = -x² (standard downward parabola) to see the effect.
Quadratic functions appear everywhere in the real world. They model projectile motion (like a basketball's arc), represent area relationships, and help find optimal solutions in business and engineering.
🚀 When you understand quadratic functions, you're actually learning the mathematics behind everything from roller coaster designs to satellite orbits!
Mastering these features gives you powerful tools to analyze graphs, solve equations, and tackle problems involving change at varying rates.
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.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Learn how to use additional information to prove the similarity of triangles using AA Similarity and solve for unknown measures.
Algebra 1A simple introduction to the basic components of a parabola and quadratic equation.
Algebra 1Features & forms of quadratic functions - Quadratics - 9th,10th,11th,12th grade high school algebra 1
Algebra 1App 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
Catalina Collazos
@catalinacollazos9
Quadratic functions are equations in the form of f(x) = ax² + bx + c that create parabola-shaped graphs. Understanding these functions helps you solve many real-world problems involving motion, area, and optimization. Let's explore the key features that make... Show more
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
A quadratic function follows the form f(x) = ax² + bx + c, where a, b, and c are constants. Each quadratic creates a parabola when graphed, with several important features to identify.
The vertex is the highest or lowest point on the parabola, found at . For example, in f(x) = 2x² - 4x + 3, the vertex sits at (1,1). This point tells you where the function reaches its maximum or minimum value.
The axis of symmetry is a vertical line passing through the vertex, dividing the parabola into mirror images. It's always at x = -b/2a. In f(x) = -x² + 6x - 5, the axis of symmetry is x = 3.
💡 Think of a quadratic function like a mirror - the axis of symmetry is where you'd place the mirror to create identical reflections on both sides!
The discriminant tells you about the roots of the equation. For f(x) = 3x² + 2x + 1, the discriminant is D = -8, which means this quadratic has no real roots .
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The vertex form of a quadratic function is f(x) = a² + k, where (h,k) is the vertex. This form makes identifying the vertex simple! For example, f(x) = 4x² + 8x + 5 can be rewritten as f(x) = 4² + 1, showing the vertex is at (-1,1).
The direction of the parabola depends on the value of a. When a > 0, the parabola opens upward (like a cup). When a < 0, it opens downward (like an inverted cup). For instance, f(x) = -x² + 2x + 1 opens downward because a = -1.
The y-intercept occurs where the parabola crosses the y-axis, always at (0,c). For f(x) = 2x² + 3x - 1, the y-intercept is (0,-1).
🔍 The sign of the leading coefficient (a) tells you everything about the parabola's direction - it's like gravity either pulling down (a > 0) or pushing up (a < 0)!
The roots or zeros are where f(x) = 0, found using the quadratic formula: x = /2a. For example, f(x) = x² - 4x + 3 has roots at x = 1 and x = 3.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Every quadratic function has either a maximum or minimum value at its vertex. When the parabola opens downward (a < 0), the vertex is a maximum. When it opens upward (a > 0), the vertex is a minimum. For f(x) = -2x² + 4x + 7, the maximum value is 9, occurring at x = 1.
The shape of the parabola depends primarily on the value of a. A larger |a| makes the parabola narrower, while a smaller |a| makes it wider. Compare f(x) = x² (standard upward parabola) with f(x) = -x² (standard downward parabola) to see the effect.
Quadratic functions appear everywhere in the real world. They model projectile motion (like a basketball's arc), represent area relationships, and help find optimal solutions in business and engineering.
🚀 When you understand quadratic functions, you're actually learning the mathematics behind everything from roller coaster designs to satellite orbits!
Mastering these features gives you powerful tools to analyze graphs, solve equations, and tackle problems involving change at varying rates.
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.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Learn how to use additional information to prove the similarity of triangles using AA Similarity and solve for unknown measures.
Algebra 1A simple introduction to the basic components of a parabola and quadratic equation.
Algebra 1Features & forms of quadratic functions - Quadratics - 9th,10th,11th,12th grade high school algebra 1
Algebra 1App 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
Calculus 1
Overige