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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Calculations data - enter values, that you know here

Function value (yy)
(the function value at single point x, often marked as f(x))
=>
Vertical asymptote of homographic function (x)
=>
Horizontal asymptote of homographic function (y)
=>
Dicriminant of the homographic function (DD)
(when D > 0 the function is increasing, when D < 0 the function is decreasing)
=>
Zero of the function (xx)
(argument for which the function has a value of zero, its a solution of f(x) = 0 equation)
=>
Value of b/x function (yy)
(the value of f(x)=b/x function for given x, parameters a,d are zero, parameter c is 1)
=>
Function argument (xx)
<=
Coefficient a (a)
<=
Coefficient b (b)
<=
Coefficient c (c)
<=
Coefficient d (d)
<=

Result: function value (yy)

Summary
Used formulaShow sourcey=a x+bc x+d y=\frac{ a~ x+ b}{ c~ x+ d}
ResultShow source11
Numerical resultShow source11
Result step by step
1Show source11+111+1\frac{1\cdot1+1}{1\cdot1+1}Multiply by one
2Show source1+111+1\frac{1+1}{1\cdot1+1}Multiply by one
3Show source1+11+1\frac{1+1}{1+1}Cancel terms
4Show source11Result
Numerical result step by step
1Show source11Result

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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