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**Graph Sketching: Asymptotes and Rational Functions**

Unit 6 Lesson #2 Asymptotes Graph Sketching: Asymptotes and Rational Functions OBJECTIVES Find limits involving infinity. Determine the asymptotes of a function’s graph.

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**Unit 6 Lesson #2 Asymptotes**

DEFINITION: A rational function is a function f that can be described by where P(x) and Q(x) are polynomials, with Q(x) not the zero polynomial. The domain of f consists of all inputs x for which Q(x) ≠ 0.

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**Unit 6 Lesson #2 Asymptotes**

A vertical asymptote is a vertical line that a function approaches, but never reaches. If a is a value for x that makes the denominator = 0, the line x = a is a vertical asymptote. The graph will never cross a vertical asymptote

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**Unit 6 Lesson #2 Asymptotes**

The line x = a is a vertical asymptote if any of the following limit statements are true: x = a ∞ x = a ∞ -∞

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**Unit 6 Lesson #2 Asymptotes**

A horizontal asymptote behaves differently. We determine what happens to the function as x approaches positive or negative infinity. The graph of the function can cross the horizontal asymptote. y = 0

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 1: Graph the function x-intercept y-intercept Does not exist does not exist does not exist There is a vertical asymptote at x = 1

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 1: Graph the function Now lets look at the one-sided limits around x = 1 1 x 1.0001 1.001 1.01 1.2 1.5 2 3 y 30 000 3000 300 15 6 ∞ 1 x -3 -1 0.5 0.9 0.99 0.999 y -0.75 -1.5 -6 -30 -300 -3 000 -∞

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 1: Graph the function ∞ y = 0 There is a horizontal asymptote at y = 0 (0, -3) x = 1 -∞

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 2: Graph the function x-intercept y-intercept

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 2: Graph the function does not exist does not exist There is a vertical asymptote at x = 3 x = 3

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**Unit 6 Lesson #2 Asymptotes**

Now lets look at the one-sided limits around x = 3 ∞ -∞

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**Unit 6 Lesson #2 Asymptotes**

Look for Horizontal Asymptotes ∞ y = 2 x = 3 -∞ There is a horizontal asymptote at y = 2

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 2: Graph the function ∞ y = 2 x = 3 -∞

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**EXAMPLE 3: Graph the function**

x- intercept y-intercept Domain: x ≠ -1, 3

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 3: cont There is a 'hole' at (-1, ¾ )

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**Unit 6 Lesson #2 Asymptotes**

EXAMPLE 3: cont DNE x = 3 There is a vertical asymptote at x = 3

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**Unit 6 Lesson #2 Asymptotes**

Look at one-sided limits around x = 3 From the left From the right f(2.999) = ∞ f(3.001) = ∞ ∞ x = 3 -∞

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**Unit 6 Lesson #2 Asymptotes**

Is there a horizontal asymptote? There is a horizontal asymptote at y = 1 ∞ y = 1 x = 3 -∞

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**Unit 6 Lesson #2 Asymptotes**

∞ x = 3 y = 1 (2, 0) - ∞

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