4. Polynomial and Rational Functions

Identify Vertical Asymptotes

Rational Functions and Their Graphs

Finding Vertical Asymptotes of Rational Functions

Zeros and Vertical Asymptotes of a Rational Function

Rational Functions of the Form y = 1/x^n and Their Transformations / Example 11.5

In Exercises 1–8, find the domain of each rational function. g(x)=3x^2/(x−5)(x+4)

Find the Equations of the Vertical and Horizontal Asymptotes of Rational Function f(x)=(3x+12)/(x+2)

Finding All Asymptotes of a Rational Function (Vertical, Horizontal, Oblique / Slant)

 General Attributes of Rational Functions

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/25b925f2c7f1c083-1675372297034.jpg" alt="" /> As x -> -3^-, f(x) -> __

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=(x+7)/(x^2+4x−21)

Rational Equations- Work-rate Problem

Arrow Notation

Comprehensive Graph of a Rational Function

Rational Functions: Vertical and Horizontal Asymptotes, Arrow notation

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/0d03926f45ada28f-1675372401938.jpg" alt="" /> As x -> -∞, f(x) -> __

Coordinates of a Hole of a Rational Function

Arrow Notation -Rational Functions

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. r(x) = (x^2 + 4x + 3)/(x + 2)^2

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=(x^2+4x−21)/(x+7)

Rational Function : Find the Asymptotes (Vertical, Horizontal, Slant / Oblique)

 Rational Functions of the Form y = 1/x^n and Their Transformations / Example 11.6

Graphing a Rational Function that has an Oblique/Slant Asymptote and  a Vertical Asymptote

Graphing a Rational Function with Slant / Oblique Asymptote

Vertical Asymptotes of Rational Functions: Quick Way to Find Them, Another Example 2

How to Find Slant Asymptote of a Rational Function

 Finding Horizontal Asymptotes Using Leading Terms

Identify a vertical asymptote

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x^3/(3x^2+1)

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/ff21d62ae02a27ee-1675372430629.jpg" alt="" /> As x -> 1^-, f(x) -> __

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=x/(x+4)

Finding Oblique Asymptotes by Long Division

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x^2+4)

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

Graphing Rational Functions and Their Asymptotes

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=(x^2−9)/(x−3)

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/02db0cd75ab33e1d-1675451421012.jpg" alt="" /> As x -> 1^+, f(x) -> __

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (4x^2 - 16x + 16)/(2x - 3)

Shortcut to Find Horizontal Asymptotes of Rational Functions

In Exercises 1–8, find the domain of each rational function. f(x)=5x/(x−4)

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x−3)/(x^2−9)

Create and graph relationships using rational functions

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

In Exercises 55–56, use transformations of f(x) = (1/x) or f(x) = (1/x^2) to graph each rational function. g(x) = 1/(x + 2)^2 - 1<img src="https://lightcat-files.s3.amazonaws.com/problem_images/e42381fd18f9909d-1672078344441.jpg" alt="" />

Rational Equations- Applications with Uniform Motion

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x+3)/x(x+4)

MATHS - Graphs - Asymptotes

Finding the Domain of Rational Functions

Using Arrow Notation

In Exercises 1–8, find the domain of each rational function. h(x)=(x+7)/(x^2−49)

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=x/x(x+4)

 Rational Functions of the Form y = 1/x^n and Their Transformations / Example 11.4

Vertical Asymptotes of Rational Functions: Quick Way to Find Them, Another Example 1

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x^2+1)

How to find SLANT ASYMPTOTES 

Finding the Domain of a Rational Function / Fraction with Variable in Denominator

Rational Expressions and Domain

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/316e217bab49da77-1675452034423.jpg" alt="" /> As x -> -2^+, f(x) -> __

In Exercises 1–8, find the domain of each rational function. f(x)=(x+7)/(x^2+49)

Finding the Domain of an Expression Involving Fractions - Example 1

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x^2/(3x^2+1)

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