Objective: In this lesson you learned how to determine the domains of rational functions, find ... The domain of a ratio
Section 2.6
•
Rational Functions
Name______________________________________________ Section 2.6 Rational Functions Objective: In this lesson you learned how to determine the domains of rational functions, find asymptotes of rational functions, and sketch the graphs of rational functions.
Important Vocabulary
Define each term or concept.
Rational function A function that can be written in the form: f(x) = N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. Vertical asymptote The line x = a is a vertical asymptote of the graph of f if f(x) → ∞ or f(x) → – ∞ as x → a , either from the right or from the left. Horizontal asymptote The line y = b is a horizontal asymptote of the graph of f if f(x) → b as x → ∞ or x → – ∞. Slant (or oblique) asymptote If the degree of the numerator of a rational function is exactly one more than the degree of the denominator, then the line determined by the quotient of the denominator into the numerator is a slant asymptote of the graph of the rational function. I. Introduction (Page 184) The domain of a rational function of x includes all real numbers except . . .
What you should learn How to find the domains of rational functions
x-values that make the denominator zero.
To find the domain of a rational function of x, . . .
set the
denominator of the rational function equal to zero and solve for x. These values of x must be excluded from the domain of the function. Example 1: Find the domain of the function f ( x) =
1
. x −9 The domain of f is all real numbers except x = − 3 and x = 3. 2
II. Horizontal and Vertical Asymptotes (Pages 185−186)
What you should learn How to find the horizontal and vertical asymptotes of graphs of rational functions
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50
Chapter 2
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Polynomial and Rational Functions
Let f be the rational function given by N ( x) a n x n + a n −1 x n −1 + L + a1 x + a 0 f ( x) = = D( x) bm x m + bm −1 x m −1 + L + b1 x + b0 where N(x) and D(x) have no common factors. 1) The graph of f has vertical asymptotes at of D(x)
the zeros
.
2) The graph of f has one or no horizontal asymptote determined by D(x)
comparing the degrees of N(x) and
.
a) If n < m, the graph of f has
the line y = 0
(the x-axis) as a horizontal asymptote b) If n = m, the graph of f has
the line
y = an/bm as a horizontal asymptote c) If n > m, the graph of f has asymptote
.
.
no horizontal
.
Example 2: Find the asymptotes of the function 2x − 1 f ( x) = 2 . x − x−6 Vertical: x = − 2, x = 3; Horizontal: y = 0
III. Analyzing Graphs of Rational Functions (Pages 187−189)
To sketch the graph of the rational function f(x) = N(x)/D(x),
What you should learn How to analyze and sketch graphs of rational functions
Name______________________________________________ 7) Use smooth curves to complete the graph between and beyond the vertical asymptotes.
Example 3: Sketch the graph of f ( x) =
3x . x+4
y
x
IV. Slant Asymptotes (Page 190)
To find the equation of a slant asymptote, . . .
use long
division to divide the denominator of the rational function into
What you should learn How to sketch graphs of rational functions that have slant asymptotes
the numerator. The equation of the slant asymptote is the quotient, excluding the remainder. Example 4: Decide whether each of the following rational functions has a slant asymptote. If so, find the equation of the slant asymptote. 3x 3 + 2 x3 − 1 (a) f ( x) = 2 (b) f ( x) = 2x − 5 x + 3x + 5 (a) Yes, y = x − 3 (b) No
V. Applications of Rational Functions (Pages 191−192)
Give an example of asymptotic behavior that occurs in real life. Answers will vary.
