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Algebra 2

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Rational Functions

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Graphs of Rational Functions

Learn Even and Odd Functions & Find Asymptotes Easily

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Learn Even and Odd Functions & Find Asymptotes Easily
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irfane

@minalovesmangoes

·

256 Followers

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A comprehensive guide to understanding even and odd functions in algebra and graphing rational functions with asymptotes, covering function symmetry, asymptotes, and graphing techniques.

  • Even functions demonstrate symmetry about the y-axis, with f(-x) = f(x)
  • Odd functions show symmetry about the origin, with f(-x) = -f(x)
  • Rational functions are quotients of two polynomials requiring careful analysis of asymptotes and intercepts
  • Key concepts include vertical asymptotes, horizontal asymptotes, and holes in graphs
  • Understanding domain restrictions and range limitations is crucial for accurate graphing

9/15/2023

74

EVEN 3 ODD FUNCTIONS fis even if f(-x) = f (x) for all x in the domain of f. f is odd if f(-x) = -f(x) for all x in the domain. of f. Symmet

View

Advanced Concepts in Rational Function Graphing

The second page delves deeper into the analysis of rational functions, particularly focusing on slant asymptotes and holes in graphs. It provides detailed explanations of domain restrictions and their implications.

Definition: Slant asymptotes occur when the numerator's degree exceeds the denominator's degree by exactly one.

Highlight: Domain restrictions coincide with vertical asymptotes, where x-values can approach but never equal these points.

Example: In the function f(x) = (x² + 4x + 4)/(x² - 4), factors that cancel out create holes, while remaining factors in the denominator create vertical asymptotes.

Quote: "When the factor cancels out → hole; when the factor does not cancel out = vertical asymptote"

EVEN 3 ODD FUNCTIONS fis even if f(-x) = f (x) for all x in the domain of f. f is odd if f(-x) = -f(x) for all x in the domain. of f. Symmet

View

Graphing Techniques and Analysis

The third page concludes with comprehensive techniques for analyzing and graphing rational functions, providing specific methods for finding various critical points and asymptotes.

Vocabulary: Horizontal asymptotes (HA) are determined by comparing the degrees of numerator and denominator polynomials.

Definition: A slant asymptote occurs when the numerator's degree exceeds the denominator's degree by exactly one.

Example: To find x-intercepts, set y=0 and solve; for y-intercepts, substitute x=0 into the function.

Highlight: The process of finding holes involves examining cancelled factors in the denominator, while horizontal asymptotes require comparison of leading term exponents.

EVEN 3 ODD FUNCTIONS fis even if f(-x) = f (x) for all x in the domain of f. f is odd if f(-x) = -f(x) for all x in the domain. of f. Symmet

View

Understanding Function Symmetry and Rational Functions

The first page introduces fundamental concepts of even and odd functions, along with the basics of rational functions and their graphing techniques. The material covers essential definitions and methods for analyzing these mathematical relationships.

Definition: An even function exhibits symmetry about the y-axis where f(-x) = f(x), while an odd function shows symmetry about the origin where f(-x) = -f(x).

Vocabulary: A rational function is expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials, and Q(x) cannot equal zero.

Highlight: To find horizontal asymptotes, compare the degrees of numerator and denominator polynomials, with three distinct cases determining the outcome.

Example: When finding y-intercepts, substitute x=0; for x-intercepts, set the numerator equal to zero and solve.

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Subjects

/

Algebra 2

/

Rational Functions

/

Graphs of Rational Functions

Learn Even and Odd Functions & Find Asymptotes Easily

user profile picture

irfane

@minalovesmangoes

·

256 Followers

Follow

A comprehensive guide to understanding even and odd functions in algebra and graphing rational functions with asymptotes, covering function symmetry, asymptotes, and graphing techniques.

  • Even functions demonstrate symmetry about the y-axis, with f(-x) = f(x)
  • Odd functions show symmetry about the origin, with f(-x) = -f(x)
  • Rational functions are quotients of two polynomials requiring careful analysis of asymptotes and intercepts
  • Key concepts include vertical asymptotes, horizontal asymptotes, and holes in graphs
  • Understanding domain restrictions and range limitations is crucial for accurate graphing

9/15/2023

74

 

10th/11th

 

Algebra 2

7

EVEN 3 ODD FUNCTIONS fis even if f(-x) = f (x) for all x in the domain of f. f is odd if f(-x) = -f(x) for all x in the domain. of f. Symmet

Advanced Concepts in Rational Function Graphing

The second page delves deeper into the analysis of rational functions, particularly focusing on slant asymptotes and holes in graphs. It provides detailed explanations of domain restrictions and their implications.

Definition: Slant asymptotes occur when the numerator's degree exceeds the denominator's degree by exactly one.

Highlight: Domain restrictions coincide with vertical asymptotes, where x-values can approach but never equal these points.

Example: In the function f(x) = (x² + 4x + 4)/(x² - 4), factors that cancel out create holes, while remaining factors in the denominator create vertical asymptotes.

Quote: "When the factor cancels out → hole; when the factor does not cancel out = vertical asymptote"

EVEN 3 ODD FUNCTIONS fis even if f(-x) = f (x) for all x in the domain of f. f is odd if f(-x) = -f(x) for all x in the domain. of f. Symmet

Graphing Techniques and Analysis

The third page concludes with comprehensive techniques for analyzing and graphing rational functions, providing specific methods for finding various critical points and asymptotes.

Vocabulary: Horizontal asymptotes (HA) are determined by comparing the degrees of numerator and denominator polynomials.

Definition: A slant asymptote occurs when the numerator's degree exceeds the denominator's degree by exactly one.

Example: To find x-intercepts, set y=0 and solve; for y-intercepts, substitute x=0 into the function.

Highlight: The process of finding holes involves examining cancelled factors in the denominator, while horizontal asymptotes require comparison of leading term exponents.

EVEN 3 ODD FUNCTIONS fis even if f(-x) = f (x) for all x in the domain of f. f is odd if f(-x) = -f(x) for all x in the domain. of f. Symmet

Understanding Function Symmetry and Rational Functions

The first page introduces fundamental concepts of even and odd functions, along with the basics of rational functions and their graphing techniques. The material covers essential definitions and methods for analyzing these mathematical relationships.

Definition: An even function exhibits symmetry about the y-axis where f(-x) = f(x), while an odd function shows symmetry about the origin where f(-x) = -f(x).

Vocabulary: A rational function is expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials, and Q(x) cannot equal zero.

Highlight: To find horizontal asymptotes, compare the degrees of numerator and denominator polynomials, with three distinct cases determining the outcome.

Example: When finding y-intercepts, substitute x=0; for x-intercepts, set the numerator equal to zero and solve.

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

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

15 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying