College Algebra Twelfth Edition Chapter 3 Polynomial and
College Algebra Twelfth Edition Chapter 3 Polynomial and Rational Functions Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 1
Section 3. 5 Rational Functions: Graphs, Applications, and Models • The Reciprocal Function • The Function • Asymptotes • Graphing Techniques • Rational Models Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 2
Rational Function (1 of 2) A function f of the form where are polynomials, with is a rational function. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 3
Rational Function (2 of 2) Some examples of rational functions are Since any values of x such that are excluded from the domain of a rational function, this type of function often has a discontinuous graph—that is, a graph that has one or more breaks in it. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 4
The Reciprocal Function (1 of 3) The simplest rational function with a variable denominator is the reciprocal function. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 5
The Reciprocal Function (2 of 3) The domain of this function is the set of all real numbers except 0. The number 0 cannot be used as a value of x, but it is helpful to find values of for some values of x very close to 0. We use the table feature of a graphing calculator to do this. The tables suggest that increases without bound as x gets closer and closer to 0, which is written in symbols as Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6
The Reciprocal Function (3 of 3) means that x (The symbol approaches 0, without necessarily ever being equal to 0. ) Since x cannot equal 0, the graph of will never intersect the vertical line x = 0. This line is called a vertical asymptote. As increases without bound, the values get closer and closer to 0, as of shown in the tables. Letting increase causes the without bound graph of to move closer and closer to the horizontal line y = 0. This line is a horizontal asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 7
Domain: Range: • decreases on the intervals Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 8
Domain: Range: • It is discontinuous at x = 0. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9
Domain: Range: • The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10
Domain: Range: • It is an odd function, and its graph is symmetric with respect to the origin. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11
Example 1: Graphing a Rational Function (1 of 4) Give the domain and the largest open Graph intervals of the domain which the function is increasing or decreasing. Solution The expression can be written as indicating that the graph may be obtained by stretching the graph of vertically by a factor of 2 and reflecting it across either the x-axis or y-axis. The x- and y-axes remain the horizontal and vertical asymptotes. The domain and range are both still Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12
Example 1: Graphing a Rational Function (2 of 4) Graph Give the domain and the largest open intervals of the domain which the function is increasing or decreasing. Solution The graph shows that is increasing on both sides of its vertical asymptote. Thus, it is increasing on Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 13
Example 1: Graphing a Rational Function (3 of 4) Give the domain and the largest Graph open intervals of the domain which the function is increasing or decreasing. Solution The expression can be written as indicating that the graph may be obtained by shifting the graph of to the left 1 unit and stretching it vertically by a factor of 2. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 14
Example 1: Graphing a Rational Function (4 of 4) Give the domain and the largest open Graph intervals of the domain which the function is increasing or decreasing. Solution The horizontal shift affects the domain, which is now The line x = − 1 is the vertical asymptote, and the line y = 0 (the x-axis) remains the horizontal asymptote. The range is still The graph shows that is decreasing on both sides of its vertical asymptote. Thus it is decreasing on Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 15
The rational function defined by also has domain We can use table feature of a graphing calculator to examine values of for some x-values close to 0. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 16
increases without The tables suggest that bound as x gets closer and closer to 0. Notice that as x approaches 0 from either side, function values are all positive and there is symmetry with respect to the y-axis. Thus, The y-axis (x = 0) is the vertical asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 17
approaches 0, As increases without bound, as suggested by the tables. Again, function values are all positive. The x-axis is the horizontal asymptote of the graph. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 18
Range: Domain: • increases on the interval and decreases on the interval Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 19
Range: Domain: • It is discontinuous at x = 0. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 20
Domain: Range: • The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 21
Domain: Range: • It is an even function, and its graph is symmetric with respect to the y-axis. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 22
Example 3: Graphing a Rational Function (1 of 3) Graph Give the domain and the largest open intervals of the domain which the function is increasing or decreasing. Solution The function is equivalent to where This indicates that the graph will be shifted 2 units to the left and 1 unit down. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 23
Example 3: Graphing a Rational Function (2 of 3) Graph Give the domain and the largest open intervals of the domain which the function is increasing or decreasing. Solution The horizontal shift affects the domain, now while the vertical shift affects the range, now Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 24
Example 3: Graphing a Rational Function (3 of 3) Give the domain and the Graph largest open intervals of the domain which the function is increasing or decreasing. Solution The vertical asymptote has equation x = − 2, and the horizontal asymptote has equation y = − 1. This function is increasing on and decreasing on Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 25
Asymptotes Let define polynomials. Consider the rational function real numbers a and b. 1. If vertical asymptote. written in lowest terms, and then the line x = a is a then the line y = b is a 2. If horizontal asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 26
Determining Asymptotes (1 of 4) To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures. 1. Vertical Asymptotes Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 27
Determining Asymptotes (2 of 4) 2. Other Asymptotes Determine any other asymptotes by considering three possibilities: (a) If the numerator has lesser degree than the denominator, then there is a horizontal asymptote y = 0 (the x-axis). Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 28
Determining Asymptotes (3 of 4) 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (b) If the numerator and denominator have the same degree, and the function is of the form then the horizontal asymptote has equation Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 29
Determining Asymptotes (4 of 4) 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 30
Note The graph of a rational function may have more than one vertical asymptote, or it may have none at all. The graph cannot intersect any vertical asymptote. There can be at most one other (nonvertical) asymptote, and the graph may intersect that asymptote, as we shall see in Example 7. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 31
Example 4: Finding Asymptotes Rational Functions (1 of 10) Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. (a) Solution To find the vertical asymptotes, set the denominator equal to 0 and solve. Zero-factor property Solve each equation. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 32
Example 4: Finding Asymptotes Rational Functions (2 of 10) The equations of the vertical asymptotes are and x = − 3. To find the equation of the horizontal asymptote, divide each term by the greatest power of x in the expression. First, multiply the factors in the denominator. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 33
Example 4: Finding Asymptotes Rational Functions (3 of 10) Now divide each term in the numerator and denominator by since 2 is the greatest power of x. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 34
Example 4: Finding Asymptotes Rational Functions (4 of 10) As increases without bound, the quotients all approach 0, and the value of approaches The line y = 0 (that is, the x-axis) is therefore the horizontal asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 35
Example 4: Finding Asymptotes Rational Functions (5 of 10) For each rational function f, find all asymptotes. (b) Solution Set the denominator x − 3 equal to 0 to find that the vertical asymptote has equation x = 3. To find the horizontal asymptote, divide each term in the rational expression by x since the greatest power of x in the expression is 1. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 36
Example 4: Finding Asymptotes Rational Functions (6 of 10) For each rational function f, find all asymptotes. (b) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 37
Example 4: Finding Asymptotes Rational Functions (7 of 10) As increases without bound, both approach 0, and approaches so the line y = 2 is the horizontal asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 38
Example 4: Finding Asymptotes Rational Functions (8 of 10) For each rational function , find all asymptotes. (c) Solution Setting the denominator x − 2 equal to 0 shows that the vertical asymptote has equation x = 2. If we divide by the greatest power of x as before ( in this case), we see that there is no horizontal asymptote because does not approach any real number as is undefined. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved since Slide - 39
Example 4: Finding Asymptotes Rational Functions (9 of 10) This happens whenever the degree of the numerator is greater than the degree of the denominator. In such cases, divide the denominator into the numerator to write the expression in another form. We use synthetic division, as shown. The result enables us to write the function as follows. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 40
Example 4: Finding Asymptotes Rational Functions (10 of 10) For very large values of is close to 0, and the graph approaches the line y = x + 2. This line is an oblique asymptote (slanted, neither vertical nor horizontal) for the graph of the function. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 41
Point of Discontinuity A rational function that is not in lowest terms often has a “hole, ” or point of discontinuity, in its graph. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 73
Example 9: Graphing a Rational Function Defined by an Expression That Is Not in Lowest Terms (1 of 2) Graph Solution The domain of this function cannot include 2. The expression should be written in lowest terms. Factor. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 74
Example 9: Graphing a Rational Function Defined by an Expression That Is Not in Lowest Terms (2 of 2) Graph Solution The graph of this function will be the same as the graph of y = x + 2 (a straight line), with the exception of the point with x-value 2. A “hole” appears in the graph at (2, 4). Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 75
- Slides: 44