Mathematical tables: homographic function formulas
Table shows common formulas related to homographic function such as discriminant of the function (ad-bc) or asymptotes formulas (vertical and horizontal).

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Various forms of function formulas

NameFormulaLegend
Exponential function in general formShow sourcey=ax y={ a}^{ x}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Exponential function with base e (often written as exp(x))Show sourceexp(x)=ex exp(x)={ e}^{ x}
  • exp(x)exp(x) - value of exponent function,
  • xx - function argument (called also independent value),
  • ee - number e (mathematical constant, base of natural logarithm).
Homographic function in general formShow sourcey=a x+bc x+d y=\frac{ a~ x+ b}{ c~ 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).
Function b/xShow sourcey=bx y=\frac{ b}{ x}
  • yy - value of b/x function (the value of f(x)=b/x function for given x, parameters a,d are zero, parameter c is 1),
  • xx - function argument (called also independent value),
  • b - coefficient b.
Linear function in slope-intercept formShow sourcey=a x+b y= a~ x+ b
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb - linear function coefficients (slope and free parameter).
Linear function in point-slope formShow sourcey=a(xx0)+y0 y=\mathrm{a}\left( x- x_0\right)+ y_0
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • x0x_0, y0y_0 - point coordinates.
Linear function in constant-slope formShow sourceyy0xx0=y1y0x1x0\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Zero of the linear function from constant-slope formShow sourcex=y0(x1x0)y1y0+x0 x=\frac{ y_0\cdot\left( x_1- x_0\right)}{ y_1- y_0}+ x_0
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Quadratic function in standard formShow sourcey=a x2+b x+c y= a~{ x}^{2}+ b~ x+ c
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).
Quadratic function in factored formShow sourcey=a(xx1) (xx2) y=\mathrm{a}\left( x- x_1\right)~\left( x- x_2\right)
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • x1x_1, x2x_2 - function zero points (arguments, for which function has value of zero, solutions of the f(x)=0 equation).
Quadratic function in vertex formShow sourcey=a(xh)2+k y={\mathrm{a}\left( x- h\right)}^{2}+ k
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • hh, kk - coordinates of the parabola vertex (at this point function reaches its local extremum).

Function asymptotes

NameFormulaLegend
Vertical asymptote of homographic functionShow sourcex=dc x=\frac{- d}{ c}
  • x - vertical asymptote of homographic function,
  • d - coefficient d,
  • c - coefficient c.
Horizontal asymptote of homographic functionShow sourcey=ac y=\frac{ a}{ c}
  • y - horizontal asymptote of homographic function,
  • a - coefficient a,
  • c - coefficient c.

Function discriminant

NameFormulaLegend
Discriminant of homographic functionShow sourceD=adbc D= a\cdot d- b\cdot c
  • DD - dicriminant of the homographic function (when D > 0 the function is increasing, when D < 0 the function is decreasing),
  • a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0).
Discriminant of the quadratic functionShow sourceΔ=b24 a c \Delta={ b}^{2}-4~ a~ c
  • Δ\Delta - dicriminant of the quadratic function,
  • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).

Zeroes of the function (roots)

NameFormulaLegend
Zero point of homographic functionShow sourcex=ba x=\frac{- b}{ a}
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • a - coefficient a,
  • b - coefficient b.
Zero of the linear functionShow sourcex=ba x=\frac{- b}{ a}
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • bb - free parameter (linear function crosses the OY axis at (0,b) point),
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation).
Zero of the linear function from point-slope formShow sourcex=x0y0a x= x_0-\frac{ y_0}{ a}
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • x0x_0, y0y_0 - point coordinates.
Zero of the linear function from constant-slope formShow sourcex=y0(x1x0)y1y0+x0 x=\frac{ y_0\cdot\left( x_1- x_0\right)}{ y_1- y_0}+ x_0
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
The first root of the quadratic functionShow sourcex1=bΔ2 a x_1=\frac{- b-\sqrt{ \Delta}}{2~ a}
  • x1x_1 - the first root of the function,
  • bb - coefficient before power of one (number just before x),
  • aa - coefficient before power of two (number just before x2),
  • Δ\Delta - dicriminant of the quadratic function.
The second root of the quadratic functionShow sourcex2=b+Δ2 a x_2=\frac{- b+\sqrt{ \Delta}}{2~ a}
  • x2x_2 - the second root of the function,
  • bb - coefficient before power of one (number just before x),
  • aa - coefficient before power of two (number just before x2),
  • Δ\Delta - dicriminant of the quadratic function.

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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