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95

Sep 17, 2023

8 pages

Cool Function Tricks: Easy Graph Shifts and Domain Tips

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Nelpz

@ficsps

A comprehensive guide to Transformations of functions in AP Pre-Calculus... Show more

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 2: Horizontal Stretches and Compressions

This page focuses on horizontal transformations and their effects on function graphs, particularly with scaling factors.

Vocabulary: Horizontal stretch occurs when |c| > 1 in fcxcx, while compression happens when |c| < 1.

Definition: For fcxcx, when c > 1, the graph compresses horizontally by a factor of 1/c.

Example: For fxx = x², f½x½x results in a horizontal stretch, while f2x2x creates a horizontal compression.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 3: Vertical Transformations

The content explores vertical transformations and introduces the concept of increasing and decreasing functions.

Definition: A function f is increasing on an interval I if fx1x₁ < fx2x₂ when x₁ < x₂ in I.

Example: For y = cfxx, when c > 1, the graph stretches vertically by a factor of c.

Highlight: Vertical transformations affect the y-coordinates of a function's graph.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 4: Function Behavior

This section details the analysis of function behavior, particularly focusing on intervals where functions increase or decrease.

Example: A function can have multiple intervals where it's increasing or decreasing:

  • Increasing on ,1∞,-1
  • Decreasing on 1,4-1,4
  • Increasing on [4,∞)

Vocabulary: Relative extrema are points where the function changes from increasing to decreasing or vice versa.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 5: Domain and Range Analysis

The page covers methods for identifying domain and range of functions from their graphs.

Definition: Domain is all possible x-values, while range is all possible y-values of a function.

Example: For the given function:

  • Domain: ,-∞,∞
  • Range: [-10,∞)

Highlight: Relative maximum and minimum points help determine the range of a function.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 6: Polynomial Functions

This section introduces polynomial functions and methods for finding their zeros.

Definition: A polynomial function has the form Pxx = anx^n + an-1x^n-1 + ... + a₁x + a₀

Example: Solving x² + 4 = 0 yields complex roots ±2i

Vocabulary: Zeros or roots are x-values where Pxx = 0

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 7: Factor Theorem

The page explains the Factor Theorem and its applications in polynomial analysis.

Definition: If xax-a is a factor of polynomial Pxx, then Paa = 0

Example: For Pxx = x² - 3x + 2, testing if x2x-2 is a factor by evaluating P22

Highlight: The Factor Theorem provides a method to verify potential factors of polynomials.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 8: Remainder Theorem

This final page covers the Remainder Theorem and its applications in polynomial division.

Definition: When a polynomial Pxx is divided by xax-a, the remainder equals Paa

Example: Solving 2x³ + 7x² - 4x² - 27x - 18 = 0 using factoring techniques

Highlight: The Remainder Theorem simplifies the process of polynomial division and factor verification.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

Page 1: Function Transformations

This page introduces fundamental concepts of function transformations using fxx = x². The content explores various ways functions can be transformed through shifts and reflections.

Definition: Function transformations are ways to manipulate the graph of a function while maintaining its basic shape.

Example: For y = fx+2x+2, the graph shifts 2 units left from the original function.

Highlight: Key transformations covered include:

  • Vertical shifts: y = fxx ± k
  • Horizontal shifts: y = fx±hx ± h
  • Reflections over x and y axes: y = -fxx and y = fx-x


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

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

 

AP Calculus AB/BC

95

Sep 17, 2023

8 pages

Cool Function Tricks: Easy Graph Shifts and Domain Tips

user profile picture

Nelpz

@ficsps

A comprehensive guide to Transformations of functions in AP Pre-Calculus, focusing on function transformations, polynomial properties, and graphical analysis.

  • The material covers essential concepts of function transformations including horizontal and vertical shifts
  • Explores polynomial functions, their properties, and methods... Show more

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 2: Horizontal Stretches and Compressions

This page focuses on horizontal transformations and their effects on function graphs, particularly with scaling factors.

Vocabulary: Horizontal stretch occurs when |c| > 1 in fcxcx, while compression happens when |c| < 1.

Definition: For fcxcx, when c > 1, the graph compresses horizontally by a factor of 1/c.

Example: For fxx = x², f½x½x results in a horizontal stretch, while f2x2x creates a horizontal compression.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 3: Vertical Transformations

The content explores vertical transformations and introduces the concept of increasing and decreasing functions.

Definition: A function f is increasing on an interval I if fx1x₁ < fx2x₂ when x₁ < x₂ in I.

Example: For y = cfxx, when c > 1, the graph stretches vertically by a factor of c.

Highlight: Vertical transformations affect the y-coordinates of a function's graph.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 4: Function Behavior

This section details the analysis of function behavior, particularly focusing on intervals where functions increase or decrease.

Example: A function can have multiple intervals where it's increasing or decreasing:

  • Increasing on ,1∞,-1
  • Decreasing on 1,4-1,4
  • Increasing on [4,∞)

Vocabulary: Relative extrema are points where the function changes from increasing to decreasing or vice versa.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 5: Domain and Range Analysis

The page covers methods for identifying domain and range of functions from their graphs.

Definition: Domain is all possible x-values, while range is all possible y-values of a function.

Example: For the given function:

  • Domain: ,-∞,∞
  • Range: [-10,∞)

Highlight: Relative maximum and minimum points help determine the range of a function.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 6: Polynomial Functions

This section introduces polynomial functions and methods for finding their zeros.

Definition: A polynomial function has the form Pxx = anx^n + an-1x^n-1 + ... + a₁x + a₀

Example: Solving x² + 4 = 0 yields complex roots ±2i

Vocabulary: Zeros or roots are x-values where Pxx = 0

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 7: Factor Theorem

The page explains the Factor Theorem and its applications in polynomial analysis.

Definition: If xax-a is a factor of polynomial Pxx, then Paa = 0

Example: For Pxx = x² - 3x + 2, testing if x2x-2 is a factor by evaluating P22

Highlight: The Factor Theorem provides a method to verify potential factors of polynomials.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 8: Remainder Theorem

This final page covers the Remainder Theorem and its applications in polynomial division.

Definition: When a polynomial Pxx is divided by xax-a, the remainder equals Paa

Example: Solving 2x³ + 7x² - 4x² - 27x - 18 = 0 using factoring techniques

Highlight: The Remainder Theorem simplifies the process of polynomial division and factor verification.

AP- Pre Calc
|y=f(x)
Given x² = f(x) describe the following.
transformations.
12y=f(x+2
ii) y=fu)-2 | [vertical shifts]]
The graph oе все is

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

Page 1: Function Transformations

This page introduces fundamental concepts of function transformations using fxx = x². The content explores various ways functions can be transformed through shifts and reflections.

Definition: Function transformations are ways to manipulate the graph of a function while maintaining its basic shape.

Example: For y = fx+2x+2, the graph shifts 2 units left from the original function.

Highlight: Key transformations covered include:

  • Vertical shifts: y = fxx ± k
  • Horizontal shifts: y = fx±hx ± h
  • Reflections over x and y axes: y = -fxx and y = fx-x

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

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