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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Symbolic algebra

ⓘ Hint: This calculator supports symbolic math. You can enter numbers, but also symbols like a, b, pi or even whole math expressions such as (a+b)/2. If you still don't sure how to make your life easier using symbolic algebra check out our another page: Symbolic calculations

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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)#

Used formulaShow sourcey=ax+bcx+dy=\frac{a \cdot x+b}{c \cdot x+d}
ResultShow source11
Numerical resultShow source11
Result step by step
1Show source11+111+1\frac{1 \cdot 1+1}{1 \cdot 1+1}Multiply by oneAny number multiplied by one (1) gives the same number: a1=1a=aa \cdot 1 = 1 \cdot a = a
2Show source1+111+1\frac{1+1}{1 \cdot 1+1}Multiply by oneAny number multiplied by one (1) gives the same number: a1=1a=aa \cdot 1 = 1 \cdot a = a
3Show source1+11+1\frac{1+1}{1+1}Simplify arithmetic-
4Show source21+1\frac{2}{1+1}Simplify arithmetic-
5Show source22\frac{\cancel{2}}{\cancel{2}}Cancel terms or fractions
  • Dividing a number by itself gives one, colloquially we say that such numbers "cancel-out": aa=1 \frac{\cancel{a}}{\cancel{a}} = 1
  • to find-out the simplest form of fraction we can divide the numerator and denominator by the greatest common divisor (GCD) of both numbers.
6Show source11ResultYour expression reduced to the simplest form known to us.
Numerical result step by step
1Show source11The original expression-
2Show source11ResultYour expression reduced to the simplest form known to us.

Some facts#

  • The homographic function is a function that can be presented in the below form:
    y=ax+bcx+dy=\frac{a \cdot x+b}{c \cdot x+d}
    • 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=adbcD=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:
    • and vertical asymptote:
  • 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:
    • 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):
    The b/x function has no zeros, and its symmetry point is origin of the coordinate system (point (0,0)).

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