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).

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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 $y=\frac{a \cdot x+b}{c \cdot x+d}$	$y$ - 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).

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:

$y=a \cdot x+b$
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.

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