Mathematical tables: linear function formulas
Tables show common formulas related to linear function such as various form of presentation (slope-intercept, point-slope, constant-slope etc.) or root formula (zero of a function).

# 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 !

# 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).

# Slope of the line

 Name Formula Legend Slope from two points lying on the line Show source$a=\frac{ y_1- y_0}{ x_1- x_0}$ $a$ - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),$x_0$, $y_0$ - coordinates of the first point,$x_1$, $y_1$ - coordinates of the second point. Slope from one point lying on the line (free parameter is needed) Show source$a=\frac{ y_0- b}{ x_0}$ $a$ - 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),$x_0$, $y_0$ - point coordinates.

# 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 linear function is a function that can be presented in the following form:
$y= a~ x+ b$
where:
• $y$ - function value (the function value at single point x, often marked as f(x)),
• $x$ - function argument (called also independent value),
• $a$, $b$ - linear function coefficients (slope and free parameter).
• The graph of the linear function is a straight line.
• Slope of a linear function defines the degree of slope of the line to the OX axis ("horizontal"). Depending on the slope value, we can distinguish three cases:
• when the slope is zero (a = 0) - the function is reduced to the constant function, its plot is a line parallel to the OX axis,
• when the slope is positive (a > 0) - the function is increasing, it's plot is a line going towards the upper right corner of the graph,
• when the slope is negative (a < 0) - the function is decreasing, its plot is a line going towards the lower right corner of the graph.
• A linear function can have one, infinitely many or no zeros (roots). This depends on the parameter values ​​a and b as follow:
• when the slope a is different from zero (a ≠ 0) - the function has exactly one root (zero point), the plot of the function crosses the OX axis one time in the point:
$x=\frac{- b}{ a}$
• when the slope a is zero, but the free parameter b is not (a = 0 and b ≠ 0) - function has no roots (zero points), it's plot does not cross the OX axis, the function is reduced to the form:
$y = b$
• if both the slope a and the free parameter b are zero (a = 0 and b = 0) - the function has infinite number of roots (zero points), it's plot coincides with the axis OX:
$y = 0$
• The linear function is a special case of the polynomial function with the order of 0 (when a = 0) or 1.