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

Beta version#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !

⌛ Loading...

Various forms of function formulas#

Name	Formula	Legend
Exponential function in general form	Show source $y=a^{x}$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - base of the exponential function.
Exponential function with base e (often written as exp(x))	Show source $exp(x)=e^{x}$	$exp(x)$ - value of exponent function, $x$ - function argument (called also independent value), $e$ - number e (mathematical constant, base of natural logarithm).
Homographic function in general form	Show source $y=\frac{a \cdot x+b}{c \cdot x+d}$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0).
Function b/x	Show source $y=\frac{b}{x}$	$y$ - 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), $x$ - function argument (called also independent value), b - coefficient b.
Linear function in slope-intercept form	Show source $y=a \cdot x+b$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ , $b$ - linear function coefficients (slope and free parameter).
Linear function in point-slope form	Show source $y=a\left(x-x_0\right)+y_0$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - slope (number that describes the direction and the steepness of the line, sometimes is called gradient), $x_0$ , $y_0$ - point coordinates.
Linear function in constant-slope form	Show source $\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $x_0$ , $y_0$ - coordinates of the first point, $x_1$ , $y_1$ - coordinates of the second point.
Zero of the linear function from constant-slope form	Show source $x=\frac{y_0 \cdot \left(x_1-x_0\right)}{y_1-y_0}+x_0$	$x$ - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation), $x_0$ , $y_0$ - coordinates of the first point, $x_1$ , $y_1$ - coordinates of the second point.
Quadratic function in standard form	Show source $y=a \cdot x^{2}+b \cdot x+c$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ , $b$ , $c$ - quadratic function coefficients (numbers just before x², x and free parameter).
Quadratic function in factored form	Show source $y=a\left(x-x_1\right) \cdot \left(x-x_2\right)$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - coefficient before power of two (number just before x²), $x_1$ , $x_2$ - function zero points (arguments, for which function has value of zero, solutions of the f(x)=0 equation).
Quadratic function in vertex form	Show source $y=a\left(x-h\right)^{2}+k$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - coefficient before power of two (number just before x²), $h$ , $k$ - coordinates of the parabola vertex (at this point function reaches its local extremum).

Function asymptotes#

Name	Formula	Legend
Vertical asymptote of homographic function	Show source $x=\frac{-d}{c}$	x - vertical asymptote of homographic function, d - coefficient d, c - coefficient c.
Horizontal asymptote of homographic function	Show source $y=\frac{a}{c}$	y - horizontal asymptote of homographic function, a - coefficient a, c - coefficient c.

Function discriminant#

Name	Formula	Legend
Discriminant of homographic function	Show source $D=a \cdot d-b \cdot c$	$D$ - 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 function	Show source $\Delta=b^{2}-4~a \cdot c$	$\Delta$ - dicriminant of the quadratic function, $a$ , $b$ , $c$ - quadratic function coefficients (numbers just before x², x and free parameter).

Zeroes of the function (roots)#

Name	Formula	Legend
Zero point of homographic function	Show source $x=\frac{-b}{a}$	$x$ - 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 function	Show source $x=\frac{-b}{a}$	$a$ - slope (number that describes the direction and the steepness of the line, sometimes is called gradient), $b$ - free parameter (linear function crosses the OY axis at (0,b) point), $x$ - 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 form	Show source $x=x_0-\frac{y_0}{a}$	$x$ - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation), $a$ - slope (number that describes the direction and the steepness of the line, sometimes is called gradient), $x_0$ , $y_0$ - point coordinates.
Zero of the linear function from constant-slope form	Show source $x=\frac{y_0 \cdot \left(x_1-x_0\right)}{y_1-y_0}+x_0$	$x$ - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation), $x_0$ , $y_0$ - coordinates of the first point, $x_1$ , $y_1$ - coordinates of the second point.
The first root of the quadratic function	Show source $x_1=\frac{-b-\sqrt{\Delta}}{2~a}$	$x_1$ - the first root of the function, $b$ - coefficient before power of one (number just before x), $a$ - coefficient before power of two (number just before x²), $\Delta$ - dicriminant of the quadratic function.
The second root of the quadratic function	Show source $x_2=\frac{-b+\sqrt{\Delta}}{2~a}$	$x_2$ - the second root of the function, $b$ - coefficient before power of one (number just before x), $a$ - coefficient before power of two (number just before x²), $\Delta$ - dicriminant of the quadratic function.

Some facts#

The homographic function is a function that can be presented in the below form:

$y=\frac{a \cdot x+b}{c \cdot x+d}$
where:
- $y$ - function value (the function value at single point x, often marked as f(x)),
- $x$ - 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:

$D=a \cdot d-b \cdot c$
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:
  
  $y=\frac{a}{c}$
- and vertical asymptote:
  
  $x=\frac{-d}{c}$
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:
  
  $x=\frac{-b}{a}$
- 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):

$y=\frac{b}{x}$
The b/x function has no zeros, and its symmetry point is origin of the coordinate system (point (0,0)).

Tags and links to this website#

Tags:

homographic_function · math_tables_homographic_function · homographic_function_formulas · homographic_asymptotes_formulas · vertical_asymptote_of_homographic_function · horizontal_asymptote_of_homographic_function · discriminant_of_homographic_function_formula · homographic_function_root_formula · zero_of_homographic_function_formula · hiperbola_asymptotes_formulas

Tags to Polish version:

funkcja_homograficzna · tablice_matematyczne_funkcja_homograficzna · wzory_na_funkcje_homograficzna · wzory_na_asymptoty_funkcji_homograficznej · wzor_na_asymptote_pozioma_funkcji_homograficznej · wzor_na_asymptote_pionowa_funkcji_homograficznej · wzor_na_wyznacznik_funkcji_homograficznej · wzor_na_pierwiastek_funcji_homograficnzej · wzor_na_miejsce_zerowe_funkcji_homograficznej · wzor_na_asymptoty_hiperboli

What tags this calculator has#

TABLE · MATH · A -> Z (all)

Permalink#

This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

https://calculla.com/homographic_function

Links to external sites (leaving Calculla?)#