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Dec 12, 2025
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Trevor
@trevk
Quadratic functions shape our understanding of countless real-world scenarios from... Show more
Quadratic functions can be written in standard form as f(x) = ax² + bx + c, where a≠0. When graphed, these functions create a parabola - a symmetric U-shaped curve. Every parabola has an axis of symmetry that divides it into mirror images, written as x = h, and a vertex that represents either the highest or lowest point.
Want to know if a parabola opens up or down? Just look at the value of 'a' in the function! When a > 0, the parabola opens up and has a minimum value at its vertex. When a < 0, it opens down and has a maximum value at its vertex.
The domain of any quadratic function includes all real numbers. The range depends on whether the parabola opens up (y ≥ minimum value) or opens down (y ≤ maximum value).
💡 Quick Check: Without graphing, you can tell that f(x) = 2x² - 5x + 2 opens up and has a minimum value at its vertex, while g(x) = 7 - 6x - 2x² opens down and has a maximum value.
The vertex is the most important point on a parabola - it's either the highest or lowest point of the function. For a quadratic function f(x) = ax² + bx + c, you can find the vertex in just a few steps.
To find the x-coordinate of the vertex, use the formula x = -b/(2a). Then, plug this x-value back into the original equation to find the y-coordinate. The vertex is the ordered pair (x, y).
Let's try an example: For y = 3x² + 6x - 18, we calculate x = -6/(2×3) = -1. When we substitute x = -1 into the original equation, we get y = 3(-1)² + 6(-1) - 18 = 3 - 6 - 18 = -21. So the vertex is (-1, -21).
🔍 Remember: The axis of symmetry always passes through the vertex, so for this example, the axis of symmetry is the vertical line x = -1.
Another useful way to write quadratic functions is in x-intercept form: y = a, where (s,0) and (t,0) are the x-intercepts of the function. This form makes it super easy to identify where the parabola crosses the x-axis!
The value of 'a' still tells us the direction of the parabola - when a > 0, it opens up, and when a < 0, it opens down. What's cool about this form is that the vertex is located halfway between the x-intercepts.
To find the x-coordinate of the vertex, calculate /2. To find the y-coordinate, substitute this x-value into the original function. You now have all the information needed to sketch the parabola!
🌟 Pro Tip: You can convert between standard form, vertex form, and x-intercept form by expanding or factoring the equation. Each form reveals different key features of the parabola at a glance!
When analyzing a quadratic function, you can identify all its key features without graphing. Let's see how this works with different forms of the function.
For f(x) = -² - 7, which is in vertex form, we can immediately tell that the vertex is (-2,-7), the parabola opens down (a < 0), and the axis of symmetry is x = -2. The maximum value is -7, the domain is all real numbers, and the range is y ≤ -7.
For functions in standard form like g(x) = 3x² + 6x - 18, calculate the vertex using the formula x = -b/(2a). With a = 3 > 0, this parabola opens up with a minimum value at its vertex (-1,-21).
When given a function in x-intercept form like h(x) = -2, we can immediately identify the x-intercepts as (-3,0) and (1,0). The vertex is halfway between these points at x = -1, with y = 8.
🔑 Key Insight: No matter which form a quadratic function is in, always identify: the direction it opens, the vertex, axis of symmetry, domain, range, and intercepts. These features give you a complete picture of the parabola!
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.
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
Trevor
@trevk
Quadratic functions shape our understanding of countless real-world scenarios from projectile motion to profit analysis. These U-shaped curves, called parabolas, have specific characteristics that help us predict their behavior and find important values like maximum heights or minimum costs.
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Quadratic functions can be written in standard form as f(x) = ax² + bx + c, where a≠0. When graphed, these functions create a parabola - a symmetric U-shaped curve. Every parabola has an axis of symmetry that divides it into mirror images, written as x = h, and a vertex that represents either the highest or lowest point.
Want to know if a parabola opens up or down? Just look at the value of 'a' in the function! When a > 0, the parabola opens up and has a minimum value at its vertex. When a < 0, it opens down and has a maximum value at its vertex.
The domain of any quadratic function includes all real numbers. The range depends on whether the parabola opens up (y ≥ minimum value) or opens down (y ≤ maximum value).
💡 Quick Check: Without graphing, you can tell that f(x) = 2x² - 5x + 2 opens up and has a minimum value at its vertex, while g(x) = 7 - 6x - 2x² opens down and has a maximum value.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The vertex is the most important point on a parabola - it's either the highest or lowest point of the function. For a quadratic function f(x) = ax² + bx + c, you can find the vertex in just a few steps.
To find the x-coordinate of the vertex, use the formula x = -b/(2a). Then, plug this x-value back into the original equation to find the y-coordinate. The vertex is the ordered pair (x, y).
Let's try an example: For y = 3x² + 6x - 18, we calculate x = -6/(2×3) = -1. When we substitute x = -1 into the original equation, we get y = 3(-1)² + 6(-1) - 18 = 3 - 6 - 18 = -21. So the vertex is (-1, -21).
🔍 Remember: The axis of symmetry always passes through the vertex, so for this example, the axis of symmetry is the vertical line x = -1.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Another useful way to write quadratic functions is in x-intercept form: y = a, where (s,0) and (t,0) are the x-intercepts of the function. This form makes it super easy to identify where the parabola crosses the x-axis!
The value of 'a' still tells us the direction of the parabola - when a > 0, it opens up, and when a < 0, it opens down. What's cool about this form is that the vertex is located halfway between the x-intercepts.
To find the x-coordinate of the vertex, calculate /2. To find the y-coordinate, substitute this x-value into the original function. You now have all the information needed to sketch the parabola!
🌟 Pro Tip: You can convert between standard form, vertex form, and x-intercept form by expanding or factoring the equation. Each form reveals different key features of the parabola at a glance!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When analyzing a quadratic function, you can identify all its key features without graphing. Let's see how this works with different forms of the function.
For f(x) = -² - 7, which is in vertex form, we can immediately tell that the vertex is (-2,-7), the parabola opens down (a < 0), and the axis of symmetry is x = -2. The maximum value is -7, the domain is all real numbers, and the range is y ≤ -7.
For functions in standard form like g(x) = 3x² + 6x - 18, calculate the vertex using the formula x = -b/(2a). With a = 3 > 0, this parabola opens up with a minimum value at its vertex (-1,-21).
When given a function in x-intercept form like h(x) = -2, we can immediately identify the x-intercepts as (-3,0) and (1,0). The vertex is halfway between these points at x = -1, with y = 8.
🔑 Key Insight: No matter which form a quadratic function is in, always identify: the direction it opens, the vertex, axis of symmetry, domain, range, and intercepts. These features give you a complete picture of the parabola!
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.
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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
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
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