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5/12/2023
Algebra 2
Radicals and Rationals
Rational exponents and radicals are fundamental concepts in algebra, crucial for solving complex equations and understanding advanced mathematical functions. This study guide covers key topics including root notation, properties of exponents, rational exponents, graphing root functions, solving radical equations, and working with rational functions. Students will learn to manipulate expressions, solve equations, and graph various types of functions using these concepts.
5/12/2023
339
This page focuses on graphing root functions and solving radical equations, providing students with practical skills for working with radical vs rational math.
Graphing root functions:
Example: For f = -√x, the domain is .
The guide provides transformation rules for graphing more complex root functions:
f = a√ + k
Where 'a' affects vertical stretch or compression, 'h' represents horizontal shift, and 'k' represents vertical shift.
Highlight: When graphing, always find the vertex and apply transformations in the correct order.
Solving radical equations:
Vocabulary: An extraneous solution is a solution that satisfies the equation algebraically but not in the original context of the problem.
The page also covers systems of equations involving radicals, emphasizing the importance of checking for extraneous solutions both algebraically and graphically.
This final page delves into rational functions and inequalities, providing students with tools to analyze and graph these complex mathematical expressions.
Definition: A rational function is a function of the form y = P/Q, where P and Q are polynomials and Q ≠ 0.
The guide explains how to graph rational functions, including:
Example: For y = a/ + k, the vertical asymptote is at x = h, and the horizontal asymptote is at y = k.
The page also covers more complex rational functions, introducing concepts such as:
Highlight: To find holes in a rational function graph, factor both numerator and denominator. Factors that cancel out indicate potential holes.
The guide concludes with a section on solving systems of rational inequalities:
This comprehensive coverage of rational functions and inequalities provides students with the tools needed to tackle complex problems in algebra and calculus.
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Rational exponents and radicals are fundamental concepts in algebra, crucial for solving complex equations and understanding advanced mathematical functions. This study guide covers key topics including root notation, properties of exponents, rational exponents, graphing root functions, solving radical equations, and... Show more
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This page focuses on graphing root functions and solving radical equations, providing students with practical skills for working with radical vs rational math.
Graphing root functions:
Example: For f = -√x, the domain is .
The guide provides transformation rules for graphing more complex root functions:
f = a√ + k
Where 'a' affects vertical stretch or compression, 'h' represents horizontal shift, and 'k' represents vertical shift.
Highlight: When graphing, always find the vertex and apply transformations in the correct order.
Solving radical equations:
Vocabulary: An extraneous solution is a solution that satisfies the equation algebraically but not in the original context of the problem.
The page also covers systems of equations involving radicals, emphasizing the importance of checking for extraneous solutions both algebraically and graphically.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This final page delves into rational functions and inequalities, providing students with tools to analyze and graph these complex mathematical expressions.
Definition: A rational function is a function of the form y = P/Q, where P and Q are polynomials and Q ≠ 0.
The guide explains how to graph rational functions, including:
Example: For y = a/ + k, the vertical asymptote is at x = h, and the horizontal asymptote is at y = k.
The page also covers more complex rational functions, introducing concepts such as:
Highlight: To find holes in a rational function graph, factor both numerator and denominator. Factors that cancel out indicate potential holes.
The guide concludes with a section on solving systems of rational inequalities:
This comprehensive coverage of rational functions and inequalities provides students with the tools needed to tackle complex problems in algebra and calculus.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page covers the basics of roots and radical notation, as well as essential properties of exponents.
Definition: A square root of a number 'a' is a value 'b' such that b² = a. Similarly, an nth root of 'a' is a value 'b' such that bⁿ = a.
Highlight: Odd roots can be applied to any real number, but even roots must be non-negative.
The page also introduces important properties of radicals and exponents, including:
Example: When simplifying expressions with radicals, using conjugates can be helpful. For instance, to rationalize the denominator of 1/, multiply both numerator and denominator by .
The properties of exponents are crucial for working with rational exponents and radicals. Some key properties include:
Vocabulary: Rational exponents are exponents that can be expressed as fractions. They provide an alternative way to represent roots.
The page concludes with an explanation of how to convert between radical form and rational exponent form, which is essential for solving radical and rational equations.
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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
