Kailangan mo bang i-master ang conic sections para sa Pre-Calculus?... Show more
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Dec 22, 2025
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Comprehensive Pre-Calculus Notes: First Quarter Guide
M
maxine
@naihhhh
1 / 7
Introduction to Conic Sections and Degenerate Conics
Ever wondered what happens when you slice through an ice cream cone? That's basically how conic sections work! A cone is a 3D shape with a pointy top (called the apex or vertex) and a circular base, connected by straight lines called generators.
When you cut through a double-napped cone (imagine two ice cream cones stuck together at their tips), you get different shapes depending on your cutting angle. Cut it horizontally and you get a single point. Slice it vertically through the middle and you get intersecting lines. Cut it at a slant and you get a single line.
These special cases are called degenerate conics because they're basically "collapsed" versions of the real conic sections. The magic happens when your cuts create actual curves instead of just points or lines!
Quick Tip: Think of degenerate conics as the "boring" cases - they're what happen when your slice doesn't create a proper curve.
Circles and Parabolas - The Basic Curves
Circles are probably the easiest conic section to understand - they're just all the points that are exactly the same distance (called the radius) from a center point. Cut a cone parallel to its base, and boom - you get a perfect circle!
The general equation for any conic section is Ax² + Bxy + Cy² + Dx + Ey + F = 0, but circles have their own special forms. The standard form ² + ² = r² tells you everything: the center is at (h,k) and the radius is r.
Now parabolas are where things get interesting! These U-shaped curves form when you slice the cone parallel to one of its edges. Every parabola has a focus (a special point inside the curve), a directrix (a line outside the curve), and a vertex (the turning point).
The coolest part? Every point on a parabola is exactly the same distance from both the focus and the directrix. Standard forms like y² = 4ax tell you which direction the parabola opens and how wide it is.
Real-world connection: Car headlights and satellite dishes use parabolic shapes to focus light and radio waves perfectly!
Understanding the Four Types of Conic Sections
Here's where everything clicks together! When you slice through that double-napped cone, you get four main types of non-degenerate conics depending on your cutting angle.
Cut parallel to the base and you get a circle - the most symmetric conic section. Slice it parallel to an edge and you create a parabola that opens infinitely in one direction.
Make your cut at an angle that's not parallel to any edge but doesn't go through both cones, and you get an ellipse - like a stretched circle. But here's the coolest part: if your cutting plane intersects both nappes of the double cone, you create a hyperbola with two separate curved branches.
Each conic section has its own personality and mathematical properties, but they're all part of this same geometric family. Understanding how they're created helps you remember their key characteristics and equations.
Memory hack: Think "Circle-Parabola-Ellipse-Hyperbola" and remember the cutting positions: base-edge-slanted-both nappes!
Parabolas and Ellipses - Getting Into the Details
Parabolas have some pretty specific rules. The standard forms depend on which direction they open: y² = 4ax opens right, y² = -4ax opens left, x² = 4ay opens up, and x² = -4ay opens down. The value 'a' tells you how far the focus is from the vertex.
When the vertex isn't at the origin, you use forms like ² = 4a. The latus rectum is a special chord that passes through the focus and helps you graph the parabola accurately.
Ellipses are like squished circles with two special points called foci inside them. The amazing property? The sum of distances from any point on the ellipse to both foci is always constant! The major axis is the longer diameter, while the minor axis is the shorter one.
For ellipses, remember that a > b always, and if a = b, you actually have a circle. The relationship c² = a² - b² connects the center-to-focus distance (c) with the semi-major (a) and semi-minor (b) axes.
Study tip: For ellipses, whichever variable has the larger denominator determines whether it's horizontal or vertical!
Ellipse Properties and Hyperbola Basics
Ellipses have tons of important parts to remember. The vertices (V1 and V2) are the endpoints of the major axis with total length 2a, while the covertices (B1 and B2) are endpoints of the minor axis with length 2b. The foci (F1 and F2) are 2c apart.
The latera recta (plural of latus rectum) pass through each focus and have length 2b²/a. Standard forms are x²/a² + y²/b² = 1 for horizontal ellipses and x²/b² + y²/a² = 1 for vertical ones.
Hyperbolas are totally different - they're sets of points where the absolute difference of distances to two foci stays constant. Unlike ellipses that are closed curves, hyperbolas have two separate branches that extend infinitely.
The transverse axis connects the vertices with length 2a, while the conjugate axis has length 2b. These create a rectangle that helps you draw the asymptotes - the diagonal lines that the hyperbola branches approach but never touch.
Key difference: Ellipses ADD distances to foci (constant sum), while hyperbolas use the absolute DIFFERENCE of those distances!
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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.
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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
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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❗️❗️⚠️
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Brad T
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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!
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Aubrey
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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
Comprehensive Pre-Calculus Notes: First Quarter Guide
M
maxine
@naihhhh
Kailangan mo bang i-master ang conic sections para sa Pre-Calculus? Think of these as different shapes you get when you slice through a cone at various angles - parang cutting a cake, but way more mathematical! These curves show up... Show more
Introduction to Conic Sections and Degenerate Conics
Ever wondered what happens when you slice through an ice cream cone? That's basically how conic sections work! A cone is a 3D shape with a pointy top (called the apex or vertex) and a circular base, connected by straight lines called generators.
When you cut through a double-napped cone (imagine two ice cream cones stuck together at their tips), you get different shapes depending on your cutting angle. Cut it horizontally and you get a single point. Slice it vertically through the middle and you get intersecting lines. Cut it at a slant and you get a single line.
These special cases are called degenerate conics because they're basically "collapsed" versions of the real conic sections. The magic happens when your cuts create actual curves instead of just points or lines!
Quick Tip: Think of degenerate conics as the "boring" cases - they're what happen when your slice doesn't create a proper curve.
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Circles and Parabolas - The Basic Curves
Circles are probably the easiest conic section to understand - they're just all the points that are exactly the same distance (called the radius) from a center point. Cut a cone parallel to its base, and boom - you get a perfect circle!
The general equation for any conic section is Ax² + Bxy + Cy² + Dx + Ey + F = 0, but circles have their own special forms. The standard form ² + ² = r² tells you everything: the center is at (h,k) and the radius is r.
Now parabolas are where things get interesting! These U-shaped curves form when you slice the cone parallel to one of its edges. Every parabola has a focus (a special point inside the curve), a directrix (a line outside the curve), and a vertex (the turning point).
The coolest part? Every point on a parabola is exactly the same distance from both the focus and the directrix. Standard forms like y² = 4ax tell you which direction the parabola opens and how wide it is.
Real-world connection: Car headlights and satellite dishes use parabolic shapes to focus light and radio waves perfectly!
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Understanding the Four Types of Conic Sections
Here's where everything clicks together! When you slice through that double-napped cone, you get four main types of non-degenerate conics depending on your cutting angle.
Cut parallel to the base and you get a circle - the most symmetric conic section. Slice it parallel to an edge and you create a parabola that opens infinitely in one direction.
Make your cut at an angle that's not parallel to any edge but doesn't go through both cones, and you get an ellipse - like a stretched circle. But here's the coolest part: if your cutting plane intersects both nappes of the double cone, you create a hyperbola with two separate curved branches.
Each conic section has its own personality and mathematical properties, but they're all part of this same geometric family. Understanding how they're created helps you remember their key characteristics and equations.
Memory hack: Think "Circle-Parabola-Ellipse-Hyperbola" and remember the cutting positions: base-edge-slanted-both nappes!
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Parabolas and Ellipses - Getting Into the Details
Parabolas have some pretty specific rules. The standard forms depend on which direction they open: y² = 4ax opens right, y² = -4ax opens left, x² = 4ay opens up, and x² = -4ay opens down. The value 'a' tells you how far the focus is from the vertex.
When the vertex isn't at the origin, you use forms like ² = 4a. The latus rectum is a special chord that passes through the focus and helps you graph the parabola accurately.
Ellipses are like squished circles with two special points called foci inside them. The amazing property? The sum of distances from any point on the ellipse to both foci is always constant! The major axis is the longer diameter, while the minor axis is the shorter one.
For ellipses, remember that a > b always, and if a = b, you actually have a circle. The relationship c² = a² - b² connects the center-to-focus distance (c) with the semi-major (a) and semi-minor (b) axes.
Study tip: For ellipses, whichever variable has the larger denominator determines whether it's horizontal or vertical!
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Ellipse Properties and Hyperbola Basics
Ellipses have tons of important parts to remember. The vertices (V1 and V2) are the endpoints of the major axis with total length 2a, while the covertices (B1 and B2) are endpoints of the minor axis with length 2b. The foci (F1 and F2) are 2c apart.
The latera recta (plural of latus rectum) pass through each focus and have length 2b²/a. Standard forms are x²/a² + y²/b² = 1 for horizontal ellipses and x²/b² + y²/a² = 1 for vertical ones.
Hyperbolas are totally different - they're sets of points where the absolute difference of distances to two foci stays constant. Unlike ellipses that are closed curves, hyperbolas have two separate branches that extend infinitely.
The transverse axis connects the vertices with length 2a, while the conjugate axis has length 2b. These create a rectangle that helps you draw the asymptotes - the diagonal lines that the hyperbola branches approach but never touch.
Key difference: Ellipses ADD distances to foci (constant sum), while hyperbolas use the absolute DIFFERENCE of those distances!
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What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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.
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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