Intermediate Algebra – Student Workbook – Second Edition 2013

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$Lesson 2a – Assessment - Scottsdale Community College - Math Blog$

Lesson 6a - Exponents and Rational Functions Mini-Lesson RATIONAL FUNCTIONS – KEY CHARACTERISTICS A RATIONAL FUNCTION is a function of the form f (x) = p(x) q(x) where p(x) and q(x) are polynomials and q(x) does not equal zero (remember that division by zero is undefined). Rational functions have similar shapes depending on the degree of the polynomials p(x) and q(x). However, the shapes are different enough that only the general characteristics are listed here and not a general graph: VERTICAL ASYMPTOTES f(x) will have VERTICAL ASYMPTOTES (can be more than one) at all input values where q(x) = 0. These asymptotes are vertical guiding lines for the graph of f(x) and f(x) will never cross over these lines. To find the Vertical Asymptotes for f(x), set q(x) = 0 and solve for x. For each x value that you find, x = that value is the equation of your vertical asymptote. Draw a dotted, vertical line on your graph that represents this equation. DOMAIN The x’s you found when solving q(x) = 0 are the values that are NOT part of the domain of f(x). To write your domain, suppose that your function denominator is x – a. When you solve x – a = 0, you get x = a. Therefore, your domain would be “all real numbers x not equal to a”. HORIZONTAL ASYMPTOTES *If f(x) has a HORIZONTAL ASYMPTOTE, it can be determined by the ratio of the highest degree terms in p(x) and q(x). This asymptote is a guiding line for large values of the function (big positive x’s and big negative x’s). To find the Horizontal Asymptote, make a fraction with only the highest degree term in p(x) in the numerator and the highest degree term in q(x) as the denominator. Reduce this fraction completely. If the fraction reduces to a number, then y = that number is your horizontal asymptote equation. If the fraction reduces to number , then y = 0 is your x horizontal asymptote equation. [Remember the degree of a polynomial is the highest power of the polynomial] *f(x) will have a HORIZONTAL ASYMPTOTE only if the degree of q(x) degree of p(x). NOTE: The information above is VERY general and probably very confusing at this point since no examples were provided. Go carefully through the next two examples and the information will make more sense. Scottsdale Community College Page 246 Intermediate Algebra
Lesson 6a - Exponents and Rational Functions Mini-Lesson Problem 4 WORKED EXAMPLE – KEY CHARACTERISTICS OF RATIONAL FUNCTIONS Graph f (x) = 2 and determine the horizontal and vertical asymptotes and the domain. x +3 To graph, let Y1 = 2 . Input Y1=2/(x+3) into your Y= list and note the use of ( ). x + 3 To find any VERTICAL ASYMPTOTES, set the denominator of to 0 and solve for x. x + 3 = 0, therefore x = -3. The equation of the vertical asymptote is x = -3. To find the DOMAIN, because x = -3 makes the denominator of f(x) equal zero, this value is not part of the domain. All other inputs are allowable. So, the domain for f(x) is “all real numbers except -3”. To find the HORIZONTAL ASYMPTOTE, make a fraction of the highest power term in the numerator (2) and the highest power term in the denominator (x). Reduce. Here is what the fraction looks like. 2 x What you are trying to find out is, what is the value of this function as x gets really big (positive) and really big (negative)? To answer this question, we need to apply a little abstract thinking. ABSTRACT THINKING In your mind, think of the very biggest positive number you can think of. What happens when you divide 2 by that number? Well, the result is going to be very, very small…effectively zero if your number is big enough. So, y = 0 is your horizontal asymptote equation as the same thing works for the biggest negative number you can think of. Putting all these things together gives the following graph with asymptotes labeled: Scottsdale Community College Page 247 Intermediate Algebra
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Scottsdale Community College Interm
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Table of Contents Lesson 1a - Intro
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Lesson 1a - Introduction to Functio
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Lesson 1b - Linear Equations In the
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Lesson 1b - Linear Equations Mini-L
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Lesson 4b - More on Logarithms This
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Intermediate Algebra – Student Workbook – Second Edition 2013

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