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#

 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$y=\frac{ a~ x+ b}{ c~ x+ d}$ $y$ - 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$y= a~ x+ b$ $y$ - 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$y=\mathrm{a}\left( x- x_0\right)+ y_0$ $y$ - 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$y= a~{ x}^{2}+ b~ x+ c$ $y$ - 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$y=\mathrm{a}\left( x- x_1\right)~\left( x- x_2\right)$ $y$ - 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$y={\mathrm{a}\left( x- h\right)}^{2}+ k$ $y$ - 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$\Delta={ b}^{2}-4~ a~ c$ $\Delta$ - 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:
$y=\frac{ a~ x+ b}{ c~ x+ d}$
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)).