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Asymptotes quiz

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  • What is an asymptote in the context of rational function graphs?
    An asymptote is a line that a graph approaches but does not touch, affecting the graph's end behavior.
  • How do you identify a vertical asymptote for a rational function?
    Set the denominator of the function (in lowest terms) equal to zero and solve for x.
  • What is the process for finding holes in the graph of a rational function?
    Set any canceled common factor equal to zero and solve for x; this value is where the hole occurs.
  • What is the horizontal asymptote of f(x) = 1/x?
    The horizontal asymptote is y = 0 because the degree of the numerator is less than the denominator.
  • How do you determine the horizontal asymptote when the degrees of the numerator and denominator are equal?
    Divide the leading coefficient of the numerator by the leading coefficient of the denominator.
  • Can a rational function have more than one vertical asymptote?
    Yes, a rational function can have multiple vertical asymptotes depending on its structure.
  • What is a removable discontinuity in a rational function graph?
    A removable discontinuity, or hole, occurs where a common factor is canceled in the function.
  • What is the vertical asymptote of f(x) = 1/(x^2 - 9)?
    The vertical asymptotes are at x = 3 and x = -3.
  • What is the horizontal asymptote for f(x) = 2x + 3 / x?
    The horizontal asymptote is y = 2, found by dividing the leading coefficients (2/1).
  • How do vertical asymptotes affect the domain of a rational function?
    Vertical asymptotes mark values excluded from the domain, where the denominator equals zero.
  • How do horizontal asymptotes affect the range of a rational function?
    Horizontal asymptotes indicate values that the function approaches as x goes to infinity or negative infinity.
  • What is the horizontal asymptote for f(x) = 4x^2 / (-x^3 - 5x + 9)?
    The horizontal asymptote is y = 0 because the degree of the numerator (2) is less than the denominator (3).
  • What is the horizontal asymptote for f(x) = 2x^2 / (3x^2 + x - 1)?
    The horizontal asymptote is y = 2/3, found by dividing the leading coefficients.
  • Can a graph of a rational function cross its horizontal asymptote?
    Yes, a graph can cross its horizontal asymptote, but it will still approach it as x goes to infinity.
  • What is the hole in the graph of f(x) = (x + 3) / (x^2 + 4x + 3)?
    There is a hole at x = -3, where the common factor x + 3 is canceled.