3.5-RationalFunctions

# 3.5-RationalFunctions - Rational Functions and Their Graphs...

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Rational Functions and Their Graphs Section 3.5 JMerrill,2005 Revised 08

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Why Should You Learn This? Rational functions are used to model and solve many problems in the business world. Some examples of real-world scenarios are: Average speed over a distance (traffic engineers) Concentration of a mixture (chemist) Average sales over time (sales manager) Average costs over time (CFO’s)
Introduction to Rational Functions What is a rational number? So just for grins, what is an irrational number? A rational function has the form ( ) ( ) ( ) p x f x q x where p and q are polynomial functions = A number that can be expressed as a fraction: A number that cannot be expressed as a fraction: , 2 π 5 , 3, 4.5 2

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Parent Function The parent function is The graph of the parent rational function looks like……………………. The graph is not continuous and has asymptotes 1 x
Transformations The parent function How does this move? 1 x 1 3 x +

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Transformations The parent function How does this move? 1 x 1 ( 3) x +
Transformations The parent function And what about this? 1 x 1 4 ( 2) x - -

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Transformations The parent function How does this move? 1 x 2 1 x
Transformations 2 1 x 2 1 2 x + 2 1 4 ( 3) x + - 2 1 ( 3) x -

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Domain Find the domain of 2 x 1 f(x) + = Denominator can’t equal 0 (it is undefined there) 2 0 2 x x + ≠ - ( 29 ( 29 Domain , 2 2, -∞ - ∪ - ∞ Think: what numbers can I put in for x????
You Do: Domain Find the domain of 2) 1)(x (x 1 - x f(x) + + = Denominator can’t equal 0 ( 29 ( 29 1 2 0 1, 2 x x x + + ≠ - - ( 29 ( 29 ( 29 Domain , 2 2, 1 1, -∞ - ∪ - - ∪ - ∞

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You Do: Domain Find the domain of = + 2 x f(x) x 1 Denominator can’t equal 0 2 2 1 0 1 x x + ≠ ≠ - ( 29 Domain , -∞ ∞
Vertical Asymptotes At the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below. 2 x 1 f(x) + = 2 x =- 2) 1)(x (x 1 - x f(x) + + = 1, 2 x x = - = - = + 2 x f(x) x 1 none

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Vertical Asymptotes The figure below shows the graph of 2 x 1 f(x) + = The equation of the vertical asymptote is 2 x =-
Vertical Asymptotes Definition: The line x = a is a vertical asymptote of the graph of f(x) if ( 29 f x → ∞ or ( 29 f x → -∞ as x approaches “a” either from the left or from the right.

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## This note was uploaded on 05/10/2009 for the course MATH 87765 taught by Professor Levitt during the Spring '09 term at New York Institute of Technology-Westbury.

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3.5-RationalFunctions - Rational Functions and Their Graphs...

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