1. Rational Functions and Models
4.6
Identify a rational function and state its domain
Identify asymptotes
Interpret asymptotes
Graph a rational function by using transformations
Graph a rational function by hand
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2. Rational Function A function f represented by
where p(x) and q(x) are polynomials and
q(x) ≠ 0, is a rational function.
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3. Rational Function
The domain of a rational function includes all real numbers except the zeros of the denominator q(x).
The graph of a rational function is continuous except at x-values where q(x) = 0.
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4. Example 1
For each rational function, determine any horizontal or vertical asymptotes. a) b) c)
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5. Example 1
Solution a) b)
Is a rational function - both numerator and denominator are polynomials; domain is all real numbers; x2 + 1 ≠ 0
Is NOT a rational function Denominator is not a polynomial; domain is
{x | x > 0}
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6. Example 1 Solution continued c)
Is a rational function - both numerator and denominator are polynomials; domain is {x | x ≠1, x ≠ 2} because (x – 1)(x – 2) = 0 when x = 1 and x = 2.
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7. Vertical Asymptotes
The line x = k is a vertical asymptote of the graph of f if f(x) g ∞ or f(x) g –∞ as x approaches k from
either the left or the right.
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8. Horizontal Asymptotes
The line y = b is a horizontal asymptote of the graph of f if
f(x) g b as x
approaches
either ∞ or –∞.
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9. Finding Vertical & Horizontal Asymptotes
Let f be a rational function given by
written in lowest terms.
Vertical Asymptote
To find a vertical asymptote, set the denominator, q(x), equal to 0 and solve. If k is a zero of q(x), then x = k is a vertical asymptote. Caution: If k is a zero of both q(x) and p(x), then f(x) is not written in lowest terms, and x – k is a common factor.
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10. Finding Vertical & Horizontal Asymptotes
(a) If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote.
(b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
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11. Finding Vertical & Horizontal Asymptotes
(c) If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.
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12. Example 4
For each rational function, determine any horizontal or vertical asymptotes.
Solution
g(x) is a translation of f(x) left one unit and down 2 units.
The vertical asymptote is x = 1
The horizontal asymptote is
y = 2
g(x) = f(x + 1)  2
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13. Example 4
For each rational function, determine any horizontal or vertical asymptotes. a) b) c)
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14. Example
Solution a)
Degree of numerator and denominator are both 1. Since the ratio of the leading coefficients is 6/3, the horizontal asymptote is y = 2.
When x = –1, the denominator, 3x + 3, equals 0 and the numerator, 6x – 1 does not equal 0, so the vertical asymptote is x = 1
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15. Example Solution continued a) Here’s a graph of f(x).
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16. Example Solution continued b)
Degree of numerator is one less than the degree of the denominator so the x-axis, or y = 0, is a horizontal asymptote.
When x = ±2, the denominator, x2 – 4, equals 0 and the numerator, x + 1 does not equal 0, so the vertical asymptotes are x = 2 and x = 2.
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17. Example Solution continued b) Here’s a graph of g(x).
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18. Example
Solution c)
Degree of numerator is greater than the degree of the denominator so there are no horizontal asymptotes.
When x = –1, both the numerator and denominator equal 0 so the expression is not in lowest terms: g(x) = x – 1, x ≠ –1. There are no vertical asymptotes.
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19. Example Solution c) Here’s the graph of h(x). A straight line with the
point (–1, –2) missing.
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20. Slant, or Oblique, Asymptotes
A third type of asymptote, which is neither vertical nor horizontal, occurs when the numerator of a rational function has degree one more than the degree of the denominator.
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21. Slant, or Oblique, Asymptotes
The line y = x + 1 is a slant asymptote, or oblique asymptote of the graph of f.
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22. Graphs and Transformations of Rational Functions
Graphs of rational functions can vary greatly in complexity.
We begin by graphing and then use transformations to graph other rational functions.
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23. Example 5 Sketch a graph of and identify any asymptotes. Solution
Vertical asymptote:
x = 0
Horizontal asymptote:
y = 0
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24. Example 6 Use the graph of to sketch a graph of Include all asymptotes
in your graph. Write
g(x) in terms of f(x).
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25. Example 6
Solution
g(x) is a translation of f(x) left 2 units and then a reflection across
the x-axis. Vertical asymptote: x = –2
Horizontal asymptote: y = 0
g(x) = –f(x + 2)
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26. Example 7 Let a) Use a calculator to graph f. Find the domain of f.
b) Identify any vertical or horizontal asymptotes.
c) Sketch a graph of f that includes the asymptotes.
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27. Example 7
Solution
a) Here’s the calculator display using “Dot Mode.” The function is undefined when x2 – 4 = 0, or when x = ±2.
The domain of f is D = {x|x ≠ 2, x ≠ –2}.
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28. Example 7 Solution b) When x = ±2, the denominator x2 – 4
= 0 (the numerator does not), so the vertical asymptotes are x = ±2.
Degree of numerator = degree of denominator, ratio of leading coefficients is 2/1 = 2, so the horizontal asymptote is y = 2.
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29. Example 7 Solution c) Here’s another version of the graph.
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30. Graphing Rational Functions by Hand
Let define a rational function
in lowest terms. To sketch its graph, follow these steps.
STEP 1: Find all vertical asymptotes.
STEP 2: Find all horizontal or oblique asymptotes.
STEP 3: Find the y-intercept, if possible, by evaluating f(0).
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31. Graphing Rational Functions by Hand
STEP 4: Find the x-intercepts, if any, by solving f(x) = 0. (These will be the zeros of the numerator p(x).)
STEP 5: Determine whether the graph will intersect its nonvertical asymptote y = b by solving f(x) = b, where b is the y-value of the horizontal asymptote, or by solving f(x) = mx + b, where y = mx + b is the equation of the oblique asymptote.
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32. Graphing Rational Functions by Hand
STEP 6: Plot selected points as necessary. Choose an x-value in each interval of the domain determined by the vertical asymptotes and x-intercepts.
STEP 7: Complete the sketch.
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33. Example 8 Graph Solution STEP 1: Vertical asymptote: x = 3
STEP 2: Horizontal asymptote: y = 2
STEP 3: f(0) = , y-intercept is
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34. Example 8 Solution continued STEP 4: Solve f(x) = 0 The x-intercept is
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35. Example 8 Solution continued
STEP 5: Graph does not intersect its horizontal asymptote, since f(x) = 2 has no solution.
STEP 6: The points are on the graph.
STEP 7: Complete the sketch (next slide)
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36. Example 8 Solution continued STEP 7
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