Subjects

Algebra 2

Factoring

Factoring Using Structure

114

•

Nov 2, 2023

•

3 pages

domain and range + porabolas

Name: Knowunity
Brand: Knowunity

Mercedes

@mercedes_wqdg

This parabola and domain/range guide offers essential insights for students... Show more

W k k l ll
-24x65 → Domain: inputs 1
DOMAIN & RANGE
Ex: X variable, represents the domain.
1≤ y ≤s → Range : outputs I
Exs y variable : сус

Hyperbolas: Key Characteristics and Graphing

This page focuses on hyperbolas, a type of conic section with unique properties and graphical representations. It provides essential information for students learning about more advanced function types.

The page begins by introducing a specific hyperbola example: y = 1/ $x+3$ - 3. This function is used to illustrate key characteristics of hyperbolas.

Highlight: Hyperbolas have asymptotes, which are lines that the graph approaches but never touches.

Several important properties of this hyperbola are discussed:

The function can never equal -3 or 0, which relates to its vertical and horizontal asymptotes.
The graph starts downward and then goes up, which can be verified by plugging in x = 0.
It has two lines of symmetry.
There is no x-intercept.

Vocabulary: Asymptotes are lines that a curve approaches as it heads towards infinity.

The domain and range of this hyperbola are also explained:

Example: For this hyperbola, the domain is all real numbers except -3, and the range is all real numbers except 0.

The page emphasizes the behavior of the function:

Highlight: This hyperbola is always decreasing, which is a characteristic of negative hyperbolas.

Finally, the asymptotes are explicitly stated:

Vertical asymptote: x = -3
Horizontal asymptote: y = 0

This concise overview provides students with a clear understanding of hyperbola characteristics, which is crucial for graphing and analyzing these functions.

Parabolas and Quadratic Functions: Understanding Transformations

This page delves into parabolas and quadratic functions, focusing on how various parameters affect their shape and position. It's an essential topic for students learning about function transformations and graphing techniques.

The page starts by introducing the parent function of a parabola: y = x². This serves as the basis for understanding all transformations.

Definition: The parent function of a parabola is y = x², which has its vertex at $0,0$ and opens upward.

The concept of "flipping" a parabola is introduced, which occurs when the coefficient 'a' is negative.

Highlight: When 'a' is negative, the parabola opens downward, effectively flipping the graph vertically.

The page then explains the three main parameters that affect a parabola's shape and position:

'a': The multiplier that affects the shape and direction of the parabola.
'k': The vertical shift of the parabola.
'h': The horizontal shift of the parabola.

Example: In the equation y = a $x-h$ ² + k, 'h' shifts the parabola horizontally, and 'k' shifts it vertically.

The effect of the 'a' parameter is explored in more detail:

Highlight: When |a| > 1, the parabola stretches vertically. When 0 < |a| < 1, the parabola compresses vertically.

An example of a compressed parabola is given: y = 0.5x² or y = $1/2$ x².

The page concludes by emphasizing that parabolas are symmetrical, with answers mirroring each other on either side of the vertex.

This comprehensive overview provides students with a solid understanding of how to manipulate and graph parabolas, which is crucial for analyzing quadratic functions.

Domain and Range: Understanding Inequalities and Functions

This page introduces the fundamental concepts of domain and range, focusing on their representation using inequalities. It provides essential information for students learning about function analysis and graphing.

The page begins by explaining the basic notation for domain and range using inequalities. Domain inequality examples are provided, showing how to express the input values of a function. Similarly, range inequality examples demonstrate how to represent the output values.

Definition: Domain refers to the set of all possible input values $x$ for a function, while range encompasses all possible output values $y$ .

Several examples are presented to illustrate different scenarios:

Example: For the domain -2 ≤ x < 4, x is greater than or equal to -2 and less than 4.

The page also covers special cases, such as discrete domains and ranges:

Example: A function with domain {-4, 1, 3} and range {1, 2, 3, 4}.

Importantly, the concept of "all real numbers" is introduced for both domain and range, represented by the infinity symbol $∞$ .

Highlight: When expressing domain and range as inequalities, use "less than or equal to" $≤$ and "greater than or equal to" $≥$ symbols to include endpoint values.

The page concludes with more complex examples, including radical and quadratic functions, demonstrating how to write domain and range as inequalities for these cases.

Example: For f $x$ = ³√x, the domain is x ≥ 0, and the range is y ≥ 0.

This comprehensive overview provides students with a solid foundation for understanding and expressing domains and ranges using inequality notation.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

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

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

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

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

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

Elisha

iOS user

Paul T

iOS user

Subjects

Algebra 2

Factoring

Factoring Using Structure

Algebra 2

•

114

•

Nov 2, 2023

•

3 pages

domain and range + porabolas

Mercedes

@mercedes_wqdg

This parabola and domain/range guide offers essential insights for students learning about quadratic functions and inequalities. It covers key concepts, notations, and graphing techniques for various mathematical functions.

• The guide explains domain and range using inequality notation, with examples... Show more

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Hyperbolas: Key Characteristics and Graphing

The page begins by introducing a specific hyperbola example: y = 1/ $x+3$ - 3. This function is used to illustrate key characteristics of hyperbolas.

Highlight: Hyperbolas have asymptotes, which are lines that the graph approaches but never touches.

Several important properties of this hyperbola are discussed:

The function can never equal -3 or 0, which relates to its vertical and horizontal asymptotes.
The graph starts downward and then goes up, which can be verified by plugging in x = 0.
It has two lines of symmetry.
There is no x-intercept.

Vocabulary: Asymptotes are lines that a curve approaches as it heads towards infinity.

The domain and range of this hyperbola are also explained:

Example: For this hyperbola, the domain is all real numbers except -3, and the range is all real numbers except 0.

The page emphasizes the behavior of the function:

Highlight: This hyperbola is always decreasing, which is a characteristic of negative hyperbolas.

Finally, the asymptotes are explicitly stated:

Vertical asymptote: x = -3
Horizontal asymptote: y = 0

This concise overview provides students with a clear understanding of hyperbola characteristics, which is crucial for graphing and analyzing these functions.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Parabolas and Quadratic Functions: Understanding Transformations

The page starts by introducing the parent function of a parabola: y = x². This serves as the basis for understanding all transformations.

Definition: The parent function of a parabola is y = x², which has its vertex at $0,0$ and opens upward.

The concept of "flipping" a parabola is introduced, which occurs when the coefficient 'a' is negative.

Highlight: When 'a' is negative, the parabola opens downward, effectively flipping the graph vertically.

The page then explains the three main parameters that affect a parabola's shape and position:

'a': The multiplier that affects the shape and direction of the parabola.
'k': The vertical shift of the parabola.
'h': The horizontal shift of the parabola.

Example: In the equation y = a $x-h$ ² + k, 'h' shifts the parabola horizontally, and 'k' shifts it vertically.

The effect of the 'a' parameter is explored in more detail:

Highlight: When |a| > 1, the parabola stretches vertically. When 0 < |a| < 1, the parabola compresses vertically.

An example of a compressed parabola is given: y = 0.5x² or y = $1/2$ x².

The page concludes by emphasizing that parabolas are symmetrical, with answers mirroring each other on either side of the vertex.

This comprehensive overview provides students with a solid understanding of how to manipulate and graph parabolas, which is crucial for analyzing quadratic functions.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Domain and Range: Understanding Inequalities and Functions

Definition: Domain refers to the set of all possible input values $x$ for a function, while range encompasses all possible output values $y$ .

Several examples are presented to illustrate different scenarios:

Example: For the domain -2 ≤ x < 4, x is greater than or equal to -2 and less than 4.

The page also covers special cases, such as discrete domains and ranges:

Example: A function with domain {-4, 1, 3} and range {1, 2, 3, 4}.

Importantly, the concept of "all real numbers" is introduced for both domain and range, represented by the infinity symbol $∞$ .

Highlight: When expressing domain and range as inequalities, use "less than or equal to" $≤$ and "greater than or equal to" $≥$ symbols to include endpoint values.

The page concludes with more complex examples, including radical and quadratic functions, demonstrating how to write domain and range as inequalities for these cases.

Example: For f $x$ = ³√x, the domain is x ≥ 0, and the range is y ≥ 0.

This comprehensive overview provides students with a solid foundation for understanding and expressing domains and ranges using inequality notation.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

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

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

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

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

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

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

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

Elisha

iOS user

Paul T

iOS user