Homographic function calculator
Calculator helpful during common operations related to homographic function such as calculating value at given point, calculating discriminant or finding out function asymptotes.

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Some facts

  • The homographic function is a function that can be presented in the below form:
    y=a x+bc x+d y=\frac{ a~ x+ b}{ c~ x+ d}
    where:
    • yy - function value (the function value at single point x, often marked as f(x)),
    • xx - function argument (called also independent value),
    • a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0).
  • The graph of the homographic function is hyperbola. To determine monotonicity of the homographic function, we can calculate its discriminant:
    D=adbc D= a\cdot d- b\cdot c
    Then, depending on the value of the discriminant, the following scenarios are possible:
    • the discriminant is negative (D < 0) - the function is decreasing,
    • the discriminant is positive (D > 0) - the function is increasing,
    • the discriminant is equal to zero (D = 0) - the function is reduced to the constant function.
  • In the general case, the homographic function has two asymptotes:
    • horizontal asymptote given by the equation:
      y=ac y=\frac{ a}{ c}
    • and vertical asymptote:
      x=dc x=\frac{- d}{ c}
  • A homogeneous function can have exactly one zero point or has no zeros at all. It depends on the coefficient b:
    • if the coefficient b is different from zero (b ≠ 0) - the homographic function has exactly one zero point, its graph intersects OX axis at the point:
      x=ba x=\frac{- b}{ a}
    • if the coefficient b equals zero (b = 0) - the homographic function has no zeros, its graph does not cross the OX axis.
  • A special case of the homographic function is the function b/x (often called a/x, in this case formal parameter b is renamed to a):
    y=bx y=\frac{ b}{ x}
    The b/x function has no zeros, and its symmetry point is origin of the coordinate system (point (0,0)).

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