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Section 5.2 – Properties of Rational Functions

Section 5.2 – Properties of Rational Functions. Defn : . Rational F unction. A function in the form: . The functions p and q are polynomials. The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.

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Section 5.2 – Properties of Rational Functions

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  1. Section 5.2 – Properties of Rational Functions Defn: Rational Function • A function in the form: • The functions p and q are polynomials. • The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.

  2. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x –4} • or • (-, -4)  (-4, )

  3. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x 2} • or • (-, 2)  (2, )

  4. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x –3, 3} • or • (-, -3)  (-3, 3)  (3, )

  5. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x –3, 5} • or • (-, -3)  (-3, 5)  (5, )

  6. Section 5.2 – Properties of Rational Functions Linear Asymptotes Lines in which a graph of a function will approach. Vertical Asymptote • A vertical asymptote exists for any value of x that makes the denominator zero AND is not a value that makes the numerator zero. • Example • A vertical asymptotes exists at x = -5. • VA:

  7. Section 5.2 – Properties of Rational Functions Asymptotes Vertical Asymptote • Example • A vertical asymptote exists at x = 4. VA: • A vertical asymptote does not exist at x = 3 as it is a value that also makes the numerator zero. • A hole exists in the graph at x = 3.

  8. Section 5.2 – Properties of Rational Functions Asymptotes Horizontal Asymptote • A horizontal asymptote exists if the largest exponents in the numerator and the denominator are equal, • or • if the largest exponent in the denominator is larger than the largest exponent in the numerator. • If the largest exponent in the denominator is equal to the largest exponent in the numerator, then the horizontal asymptote is equal to the ratio of the coefficients. • If the largest exponent in the denominator is larger than the largest exponent in the numerator, then the horizontal asymptote is .

  9. Section 5.2 – Properties of Rational Functions Asymptotes Horizontal Asymptote • Example • HA: • A horizontal asymptote exists at y = 5/2. • A horizontal asymptote exists at y = 0. • HA:

  10. Section 5.2 – Properties of Rational Functions Asymptotes Oblique (slant) Asymptote • An oblique asymptote exists if the largest exponent in the numerator is one degree larger than the largest exponent in the denominator. **Note** • Other non-linear asymptotes can exist for a rational function.

  11. Section 5.2 – Properties of Rational Functions Asymptotes Oblique Asymptote • Example • An oblique asymptote exists. • Long division is required. • An oblique asymptote exists at y = x. OA:

  12. Section 5.2 – Properties of Rational Functions Asymptotes Oblique Asymptote • Example • An oblique asymptote exists. • Long division is required. • An oblique asymptote exists at y = 2x • OA:

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