Chapter Polynomial and Rational Functions.6 Rational Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1
Objectives: Find the domains of rational functions. Use arrow notation. Identify vertical asymptotes. Identify horizontal asymptotes. Use transformations to graph rational functions. Graph rational functions. Identify slant asymptotes. Solve applied problems involving rational functions. Copyright 014, 010, 007 Pearson Education, Inc.
Rational Functions Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as px ( ) f( x) qx ( ) where p and q are polynomial functions and qx ( ) 0. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. Copyright 014, 010, 007 Pearson Education, Inc. 3
Example: Finding the Domain of a Rational Function Find the domain of the rational function: x 5 f( x). x 5 Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. x 5 0 x 5 The domain of f consists of all real numbers except 5. Copyright 014, 010, 007 Pearson Education, Inc. 4
Example: Finding the Domain of a Rational Function (continued) Find the domain of the rational function: x 5 f( x). x 5 The domain of f consists of all real numbers except 5. We can express the domain in set-builder or interval notation. Domain of f = xx 5 Domain of f = (,5) (5, ) Copyright 014, 010, 007 Pearson Education, Inc. 5
Example: Finding the Domain of a Rational Function Find the domain of the rational function: x gx ( ). x 5 Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. x 5 0 x 5 x 5 The domain of g consists of all real numbers except 5 or 5. Copyright 014, 010, 007 Pearson Education, Inc. 6
Example: Finding the Domain of a Rational Function (continued) Find the domain of the rational function: x gx ( ). x 5 The domain of g consists of all real numbers except 5 or 5. We can express the domain in set-builder or interval notation. Domain of g = xx 5, x 5 Domain of g = (, 5) ( 5,5) (5, ) Copyright 014, 010, 007 Pearson Education, Inc. 7
Example: Finding the Domain of a Rational Function Find the domain of the rational function: x 5 hx ( ). x 5 Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. x 5 0 x 5 x 5i No real numbers cause the denominator of h(x) to equal 0. The domain of h consists of all real numbers. Copyright 014, 010, 007 Pearson Education, Inc. 8
Example: Finding the Domain of a Rational Function (continued) Find the domain of the rational function: x 5 hx ( ). x 5 The domain of h consists of all real numbers. We can write the domain in interval notation: Domain of h = (, ) Copyright 014, 010, 007 Pearson Education, Inc. 9
Arrow Notation We use arrow notation to describe the behavior of some functions. Copyright 014, 010, 007 Pearson Education, Inc. 10
Definition of a Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a. Copyright 014, 010, 007 Pearson Education, Inc. 11
Definition of a Vertical Asymptote (continued) The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a. Copyright 014, 010, 007 Pearson Education, Inc. 1
Locating Vertical Asymptotes px ( ) If f( x) is a rational function in which p(x) qx ( ) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f. Copyright 014, 010, 007 Pearson Education, Inc. 13
Example: Finding the Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of the rational function: x f( x). x 1 There are no common factors in the numerator and the denominator. x 1 0 x 1 x 1 The zeros of the denominator are 1 and 1. Thus, the lines x = 1 and x = 1 are the vertical asymptotes for the graph of f. Copyright 014, 010, 007 Pearson Education, Inc. 14
Example: Finding the Vertical Asymptotes of a Rational Function (continued) Find the vertical asymptotes, if any, of the graph of the rational function: x f( x). The zeros of the x 1 denominator are 1 and 1. Thus, x 1 the lines x = 1 and x = 1 are the vertical x 1 asymptotes for the graph of f. Copyright 014, 010, 007 Pearson Education, Inc. 15
Example: Finding the Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of the rational function: x 1 hx ( ). x 1 We cannot factor the denominator of h(x) over the real numbers. The denominator has no real zeros. Thus, the graph of h has no vertical asymptotes. Copyright 014, 010, 007 Pearson Education, Inc. 16
Definition of a Horizontal Asymptote Copyright 014, 010, 007 Pearson Education, Inc. 17
Locating Horizontal Asymptotes Copyright 014, 010, 007 Pearson Education, Inc. 18
Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: 9x f( x). 3x 1 The degree of the numerator,, is equal to the degree of the denominator,. The leading coefficients of the numerator and the denominator, 9 and 3, are used to obtain the equation of the horizontal asymptote. 9 The equation of the horizontal asymptote is y 3 or y = 3. Copyright 014, 010, 007 Pearson Education, Inc. 19
Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: 9x f( x). 3x 1 y 3 The equation of the horizontal asymptote is y = 3. Copyright 014, 010, 007 Pearson Education, Inc. 0
Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: 9x gx ( ). 3x 1 The degree of the numerator, 1, is less than the degree of the denominator,. Thus, the graph of g has the x-axis as a horizontal asymptote. The equation of the horizontal asymptote is y = 0. Copyright 014, 010, 007 Pearson Education, Inc. 1
Example: Finding the Horizontal Asymptote of a Rational Function (continued) Find the horizontal asymptote, if any, of the graph of the rational function: 9x gx ( ). 3x 1 The equation of the horizontal asymptote is y = 0. y 0 Copyright 014, 010, 007 Pearson Education, Inc.
Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of 3 the rational function: 9x hx ( ). 3x 1 The degree of the numerator, 3, is greater than the degree of the denominator,. Thus the graph of h has no horizontal asymptote. Copyright 014, 010, 007 Pearson Education, Inc. 3
Example: Finding the Horizontal Asymptote of a Rational Function (continued) Find the horizontal asymptote, if any, of the graph of 3 the rational function: 9x hx ( ). 3x 1 The graph has no horizontal asymptote. Copyright 014, 010, 007 Pearson Education, Inc. 4
Basic Reciprocal Functions Copyright 014, 010, 007 Pearson Education, Inc. 5
Example: Using Transformations to Graph a Rational Function 1 Use the graph of f( x) to graph x Begin with f( x) 1 x We ve identified two points and the asymptotes. Vertical asymptote x = 0 ( 1,1) 1 gx ( ) 1. x (1,1) Horizontal asymptote y = 0 Copyright 014, 010, 007 Pearson Education, Inc. 6
Example: Using Transformations to Graph a Rational Function (continued) 1 Use the graph of f( x) to graph x 1 gx ( ) 1. x The graph will shift units to the left. Subtract from each x-coordinate. The vertical asymptote is now x = Vertical asymptote x = ( 3, 1) ( 1,1) Horizontal asymptote y = 0 Copyright 014, 010, 007 Pearson Education, Inc. 7
Example: Using Transformations to Graph a Rational Function (continued) 1 Use the graph of f( x) to graph x 1 gx ( ) 1. x The graph will shift 1 unit down. Subtract 1 from each y-coordinate. The horizontal asymptote is now y = 1. Vertical asymptote x = ( 3, ) ( 1,0) Horizontal asymptote y = 1 Copyright 014, 010, 007 Pearson Education, Inc. 8
Strategy for Graphing a Rational Function The following strategy can be used to graph ( ) f( x) px. qx ( ) where p and q are polynomial functions with no common factors. 1. Determine whether the graph of f has symmetry. f ( x) f( x) y-axis symmetry f ( x) f( x) origin symmetry. Find the y-intercept (if there is one) by evaluating f(0). Copyright 014, 010, 007 Pearson Education, Inc. 9
Strategy for Graphing a Rational Function (continued) The following strategy can be used to graph ( ) f( x) px. qx ( ) where p and q are polynomial functions with no common factors. 3. Find the x-intercepts (if there are any) by solving the equation p(x) = 0. 4. Find any vertical asymptote(s) by solving the equation q(x) = 0. Copyright 014, 010, 007 Pearson Education, Inc. 30
Strategy for Graphing a Rational Function (continued) px ( ) The following strategy can be used to graph f( x) qx ( ) where p and q are polynomial functions with no common factors. 5. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. Copyright 014, 010, 007 Pearson Education, Inc. 31
Strategy for Graphing a Rational Function (continued) px ( ) The following strategy can be used to graph f( x) qx ( ) where p and q are polynomial functions with no common factors. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes. Copyright 014, 010, 007 Pearson Education, Inc. 3
Example: Graphing a Rational Function Graph: 3x 3 f( x). x Step 1 Determine symmetry. 3( x) 3 3x 3 f( x) ( x) x Because f( x) does not equal either f(x) or f(x), the graph has neither y-axis symmetry nor origin symmetry. Copyright 014, 010, 007 Pearson Education, Inc. 33
Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step Find the y-intercept. Evaluate f(0). 3(0) 3 3 f (0) 0 Step 3 Find x-intercepts. 3x 3 0 3x 3 x 1 Copyright 014, 010, 007 Pearson Education, Inc. 34
Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step 4 Find the vertical asymptote(s). x 0 x Step 5 Find the horizontal asymptote. The numerator and denominator of f have the same degree, 1. The horizontal asymptote is 3 y 3. 1 Copyright 014, 010, 007 Pearson Education, Inc. 35
Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step 6 Plot points between and beyond each x-intercept and vertical asymptotes. The x-intercept is (1,0) The y-intercept is 3 0, x 1 We will evaluate f at x Copyright 014, 010, 007 Pearson Education, Inc. 36 1 x 3 x 5 x 3 vertical asymptote x =
Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step 6 (continued) We evaluate f at the selected points. x 1 1 3 5 3 f( x) 3x 3 x 1 3 9 6 Copyright 014, 010, 007 Pearson Education, Inc. 37
Example: Graphing a Rational Function (continued) 3x 3 Graph: f( x). x Step 7 Graph the function vertical asymptote x = horizontal asymptote y = 3 Copyright 014, 010, 007 Pearson Education, Inc. 38
Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. px ( ) In general, if f( x), qx ( ) p and q have no common factors, and the degree of p is one greater than the degree of q, find the slant asymptotes by dividing q(x) into p(x). Copyright 014, 010, 007 Pearson Education, Inc. 39
Example: Finding the Slant Asymptotes of a Rational Function Find the slant asymptote of x 5x 7 f( x). x The degree of the numerator,, is exactly one more than the degree of the denominator, 1. To find the equation of the slant asymptote, divide the numerator by the denominator. 5 7 5 4 (x 5x 7) ( x ) x 1 x 1 5 Copyright 014, 010, 007 Pearson Education, Inc. 40
Example: Finding the Slant Asymptotes of a Rational Function Find the slant asymptote of (x 5x 7) ( x ) x 1 x The equation of the vertical slant asymptote is asymptote x = y = x 1. The graph of f(x) slant asymptote is shown. y = x 1 5 x 5x 7 f( x). x Copyright 014, 010, 007 Pearson Education, Inc. 41
Example: Application A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Write the cost function, C, of producing x wheelchairs. Cx ( ) 500,000 400x Copyright 014, 010, 007 Pearson Education, Inc. 4
Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Write the average cost function, C, of producing x wheelchairs. 500,000 400x Cx ( ) x Copyright 014, 010, 007 Pearson Education, Inc. 43
Example: Application A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find and interpret C(1000). 500,000 400x Cx ( ) x 500,000 400(1000) C(1000) 1000 The average cost per wheelchair of producing 1000 wheelchairs per month is $900. 900 Copyright 014, 010, 007 Pearson Education, Inc. 44
Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find and interpret C(10,000). 500,000 400x Cx ( ) x 500,000 400(10,000) C(10,000) 450 10,000 The average cost per wheelchair of producing 10,000 wheelchairs per month is $450. Copyright 014, 010, 007 Pearson Education, Inc. 45
Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find and interpret C(100,000). 500,000 400x Cx ( ) x 500,000 400(100,000) C(100,000) 100,000 The average cost per wheelchair of producing 100,000 wheelchairs per month is $405. 405 Copyright 014, 010, 007 Pearson Education, Inc. 46
Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find the horizontal asymptote for the average cost function, 500,000 400x Cx ( ). The horizontal x asymptote 500,000 400x 400x 500,000 Cx ( ) is y = 400 x x Copyright 014, 010, 007 Pearson Education, Inc. 47
Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. The average cost function is 500,000 400x Cx ( ) x The horizontal asymptote for this function is y = 400. Describe what this represents for the company. The cost per wheelchair approaches 400 as more wheelchairs are produced. Copyright 014, 010, 007 Pearson Education, Inc. 48