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Horizontal Asymptotes in Rational Functions

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Terms in this set (20)

  • What is the general form of a rational function f(x)?
    f(x) = \(\frac{a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0}{b_mx^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0}\) where n and m are the degrees of numerator and denominator.
  • What is the horizontal asymptote if degree of numerator n < degree of denominator m?
    The horizontal asymptote is the x-axis, or y = 0.
  • What is the horizontal asymptote if degree of numerator n = degree of denominator m?
    The horizontal asymptote is the line \(y=\frac{a_n}{b_m}\), the ratio of leading coefficients.
  • What happens if degree of numerator n > degree of denominator m?
    The graph has no horizontal asymptote.
  • How do you find the horizontal asymptote of a rational function?
    Compare degrees n and m of numerator and denominator and apply the rules for n < m, n = m, or n > m.
  • What does the horizontal asymptote represent on the graph?
    It represents the value that f(x) approaches as x approaches ±∞.
  • If f(x) = (3x^2 + 5) / (2x^2 - 1), what is the horizontal asymptote?
    Since n = m = 2, horizontal asymptote is y = 3/2.
  • If f(x) = (x + 1) / (x^2 + 4), what is the horizontal asymptote?
    Since n = 1 < m = 2, horizontal asymptote is y = 0.
  • If f(x) = (x^3 + 2) / (x^2 + 1), what is the horizontal asymptote?
    Since n = 3 > m = 2, there is no horizontal asymptote.
  • What is the significance of the leading coefficients in horizontal asymptotes?
    They determine the horizontal asymptote when degrees of numerator and denominator are equal.
  • How does the graph behave near the vertical asymptote compared to the horizontal asymptote?
    Near vertical asymptotes, f(x) tends to ±∞; near horizontal asymptotes, f(x) approaches a finite value.
  • What is the horizontal asymptote of f(x) = (5x) / (3x + 6)?
    Since n = m = 1, horizontal asymptote is y = 5/3.
  • What is the horizontal asymptote of f(x) = (4x^2) / (2x^2 + 1)?
    Since n = m = 2, horizontal asymptote is y = 4/2 = 2.
  • What is the horizontal asymptote of f(x) = (20x) / (5x^2 + 1)?
    Since n = 1 < m = 2, horizontal asymptote is y = 0.
  • What is the horizontal asymptote of f(x) = (15x^3) / (3x^2 + 1)?
    Since n = 3 > m = 2, no horizontal asymptote.
  • How to interpret the graph when horizontal asymptote is y = 0?
    The function values approach zero as x goes to ±∞.
  • What does it mean graphically if there is no horizontal asymptote?
    The function grows without bound or decreases without bound as x approaches infinity.
  • How does the horizontal asymptote relate to end behavior of rational functions?
    It describes the end behavior of the function as x approaches ±∞.
  • If the numerator degree is less than denominator degree, what is the limit of f(x) as x approaches infinity?
    The limit is 0, corresponding to the horizontal asymptote y = 0.
  • If the numerator and denominator have the same degree, what is the limit of f(x) as x approaches infinity?
    The limit is the ratio of leading coefficients, the horizontal asymptote.