2.6 Rational Functions
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1 2.6 Rational Functions Pre-Calculus Mr. Niedert Pre-Calculus 2.6 Rational Functions Mr. Niedert 1 / 20
2 2.6 Rational Functions 1 Rational Functions Pre-Calculus 2.6 Rational Functions Mr. Niedert 2 / 20
3 2.6 Rational Functions 1 Rational Functions 2 Horizontal and Vertical Asymptotes Pre-Calculus 2.6 Rational Functions Mr. Niedert 2 / 20
4 2.6 Rational Functions 1 Rational Functions 2 Horizontal and Vertical Asymptotes 3 Oblique/Slant Asymptotes Pre-Calculus 2.6 Rational Functions Mr. Niedert 2 / 20
5 Definition of a Rational Function A rational function can be written in the form f (x) = N(x) D(x) where N(x) and D(x) are each polynomials and D(x) is not the zero polynomial. Pre-Calculus 2.6 Rational Functions Mr. Niedert 3 / 20
6 The Domain of Rational Functions In general, the domain of a rational function of x includes all real numbers except x-values that make the denominator zero. Pre-Calculus 2.6 Rational Functions Mr. Niedert 4 / 20
7 Finding the Domain of a Rational Function Example Find the domain of f (x) = 3x. Express your answer in interval x 1 notation. Then describe the behavior of the function f near any excluded x-values. Pre-Calculus 2.6 Rational Functions Mr. Niedert 5 / 20
8 Finding the Domain of a Rational Function Practice 5 Find the domain of f (x) = x 2. Express your answer in interval 4x + 3 notation. Then describe the behavior of the function f near any excluded x-values. Pre-Calculus 2.6 Rational Functions Mr. Niedert 6 / 20
9 Exit Slip Exit Slip Find the domain of each rational function. Express your answer in interval notation. 5x a x + 2 3x 2 b 1 + 3x 8 c x 2 10x + 24 d x 2 + x 2 x Pre-Calculus 2.6 Rational Functions Mr. Niedert 7 / 20
10 Vertical Asymptotes Yesterday, we looked at the polynomial y = of the equation. 3x. Below is a graph x 1 Pre-Calculus 2.6 Rational Functions Mr. Niedert 8 / 20
11 Vertical Asymptotes Yesterday, we looked at the polynomial y = of the equation. 3x. Below is a graph x 1 Notice that f (x) decreases without bound as x approaches 1 from the left. In addition, f (x) increases without bound as x approaches 1 from the right. Pre-Calculus 2.6 Rational Functions Mr. Niedert 8 / 20
12 Vertical Asymptotes Yesterday, we looked at the polynomial y = of the equation. 3x. Below is a graph x 1 Notice that f (x) decreases without bound as x approaches 1 from the left. In addition, f (x) increases without bound as x approaches 1 from the right. This can be denoted as follows. f (x) as x 1 f (x) as x 1 + Pre-Calculus 2.6 Rational Functions Mr. Niedert 8 / 20
13 Vertical Asymptotes Yesterday, we looked at the polynomial y = of the equation. 3x. Below is a graph x 1 Notice that f (x) decreases without bound as x approaches 1 from the left. In addition, f (x) increases without bound as x approaches 1 from the right. This can be denoted as follows. f (x) as x 1 f (x) as x 1 + This then means that x = 1 is a vertical asymptote of the graph of y = 3x x 1. Pre-Calculus 2.6 Rational Functions Mr. Niedert 8 / 20
14 Horizontal Asymptotes Again, let s continue to consider the polynomial y = 3x x 1. Pre-Calculus 2.6 Rational Functions Mr. Niedert 9 / 20
15 Horizontal Asymptotes Again, let s continue to consider the polynomial y = 3x x 1. Notice that f (x) approaches 3 as x decreases without bound and f (x) approaches 3 as x increases without bound. Pre-Calculus 2.6 Rational Functions Mr. Niedert 9 / 20
16 Horizontal Asymptotes Again, let s continue to consider the polynomial y = 3x x 1. Notice that f (x) approaches 3 as x decreases without bound and f (x) approaches 3 as x increases without bound. This can be denoted as follows. f (x) 3 as x f (x) 3 as x Pre-Calculus 2.6 Rational Functions Mr. Niedert 9 / 20
17 Horizontal Asymptotes Again, let s continue to consider the polynomial y = 3x x 1. Notice that f (x) approaches 3 as x decreases without bound and f (x) approaches 3 as x increases without bound. This can be denoted as follows. f (x) 3 as x f (x) 3 as x This then means that y = 3 is a horizontal asymptote of the graph of y = 3x x 1. Pre-Calculus 2.6 Rational Functions Mr. Niedert 9 / 20
18 Asymptotes of a Rational Function Asymptotes of a Rational Function Let f be the rational function given by f (x) = N(x) D(x) where N(x) and D(x) are each polynomials and have no common factors. The graph of f has vertical asymptotes at the zeros of D(x). Pre-Calculus 2.6 Rational Functions Mr. Niedert 10 / 20
19 Asymptotes of a Rational Function Asymptotes of a Rational Function Let f be the rational function given by f (x) = N(x) D(x) where N(x) and D(x) are each polynomials and have no common factors. The graph of f has vertical asymptotes at the zeros of D(x). The graph of f has one or zero horizontal asymptotes, which we can determine by comparing the degrees of N(x) and D(x). For the sake of explanation, let n represent the degree of N(x) and d represent the degree of D(x). Pre-Calculus 2.6 Rational Functions Mr. Niedert 10 / 20
20 Asymptotes of a Rational Function Asymptotes of a Rational Function Let f be the rational function given by f (x) = N(x) D(x) where N(x) and D(x) are each polynomials and have no common factors. The graph of f has vertical asymptotes at the zeros of D(x). The graph of f has one or zero horizontal asymptotes, which we can determine by comparing the degrees of N(x) and D(x). For the sake of explanation, let n represent the degree of N(x) and d represent the degree of D(x). n < d = the graph of f has the line y = 0 (the x-axis) as a horizontal asymptote. Pre-Calculus 2.6 Rational Functions Mr. Niedert 10 / 20
21 Asymptotes of a Rational Function Asymptotes of a Rational Function Let f be the rational function given by f (x) = N(x) D(x) where N(x) and D(x) are each polynomials and have no common factors. The graph of f has vertical asymptotes at the zeros of D(x). The graph of f has one or zero horizontal asymptotes, which we can determine by comparing the degrees of N(x) and D(x). For the sake of explanation, let n represent the degree of N(x) and d represent the degree of D(x). n < d = the graph of f has the line y = 0 (the x-axis) as a horizontal asymptote. n = d = the graph of f has the line y = a n b d where a n and b d are the leading coefficients of N(x) and D(x), respectively. Pre-Calculus 2.6 Rational Functions Mr. Niedert 10 / 20
22 Asymptotes of a Rational Function Asymptotes of a Rational Function Let f be the rational function given by f (x) = N(x) D(x) where N(x) and D(x) are each polynomials and have no common factors. The graph of f has vertical asymptotes at the zeros of D(x). The graph of f has one or zero horizontal asymptotes, which we can determine by comparing the degrees of N(x) and D(x). For the sake of explanation, let n represent the degree of N(x) and d represent the degree of D(x). n < d = the graph of f has the line y = 0 (the x-axis) as a horizontal asymptote. n = d = the graph of f has the line y = a n b d where a n and b d are the leading coefficients of N(x) and D(x), respectively. n > d = the graph of f has no horizontal asymptote. Pre-Calculus 2.6 Rational Functions Mr. Niedert 10 / 20
23 Finding Horizontal and Vertical Asymptotes Example Find all horizontal and vertical asymptotes of the graph of f (x) = 2x 2 7x + 3 x 2 5x + 6. Pre-Calculus 2.6 Rational Functions Mr. Niedert 11 / 20
24 Finding Horizontal and Vertical Asymptotes Practice Find all horizontal and vertical asymptotes of the graph of each rational function below. a f (x) = 2x 2 x 2 1 b f (x) = 5 x 2 c f (x) = x 2 + x 2 x 2 x 6 d f (x) = x 2 x 4 Pre-Calculus 2.6 Rational Functions Mr. Niedert 12 / 20
25 Exit Slip Exit Slip Find all horizontal and vertical asymptotes of the graph of each rational function below. a f (x) = x 3 x 2 4 x 6 b f (x) = x 2 8x + 12 c f (x) = x 2 6x + 9 x 2 7x + 10 Pre-Calculus 2.6 Rational Functions Mr. Niedert 13 / 20
26 Rational Functions Assignment 1 Due Next Class: pg. 193 #5-16 Pre-Calculus 2.6 Rational Functions Mr. Niedert 14 / 20
27 Asymptotes Discovery Look at each of the graphs below. Just from the graphs, estimate where there might be asymptotes. Graph 1 Graph 2 Graph 3 Pre-Calculus 2.6 Rational Functions Mr. Niedert 15 / 20
28 Oblique/Slant Asymptotes As we saw on the previous slide, Graph 3 has a oblique (or slant) asymptote. Pre-Calculus 2.6 Rational Functions Mr. Niedert 16 / 20
29 Oblique/Slant Asymptotes As we saw on the previous slide, Graph 3 has a oblique (or slant) asymptote. Graph 3 is the graph of the function f (x) = x 2 x x + 1. It has an oblique/slant asymptote because the degree of the numerator (n) is exactly one degree greater than the degree of the denominator (d). Pre-Calculus 2.6 Rational Functions Mr. Niedert 16 / 20
30 A Rational Function with a Slant Asymptote Example Find all of the asymptotes (vertical, horizontal, and/or oblique/slant) of the function f (x) = x 2 x 2. x 1 Pre-Calculus 2.6 Rational Functions Mr. Niedert 17 / 20
31 A Rational Function with a Slant Asymptote Practice Find all of the asymptotes of the function f (x) = 3x x Pre-Calculus 2.6 Rational Functions Mr. Niedert 18 / 20
32 Exit Slip Exit Slip The asymptotes of f (x) = x 2 x (from Graph 3 earlier) were at x = 1 x + 1 and y = x 2. Describe and/or show why this is the case. Pre-Calculus 2.6 Rational Functions Mr. Niedert 19 / 20
33 Rational Functions Assignment 2 Due Next Class: Rational Functions Assignment 2 Worksheet This is the end of the section, so I should have Assignment 1 (pg. 193 #5-16) by the next class as well. Pre-Calculus 2.6 Rational Functions Mr. Niedert 20 / 20
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