Tables show common formulas related to quadratic function such as various form of representation (standard, factored, vertex etc.) or formula for discriminant of quadratic function often called simply delta

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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).

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.

Parabola vertex#

Name	Formula	Legend
The x coordinate of parabola vertex	Show source $h=\frac{-b}{2~a}$	$h$ - x coordinate of the parabola vertex (for this argument the function reaches its local extremum), $b$ - coefficient before power of one (number just before x), $a$ - coefficient before power of two (number just before x²).
The y coordinate of parabola vertex	Show source $k=\frac{-\Delta}{4~a}$	$k$ - y coordinate of the parabola vertex, $\Delta$ - dicriminant of the quadratic function, $a$ - coefficient before power of two (number just before x²).

Some facts#

The quadratic function is a function that can be prepresented in the form:

$y=a \cdot x^{2}+b \cdot x+c$
where:
- $y$ - function value (the function value at single point x, often marked as f(x)),
- $x$ - function argument (called also independent value),
- $a$ , $b$ , $c$ - quadratic function coefficients (numbers just before x², x and free parameter).
The graph of the quadratic function is parabola. Depending on the coefficient value at the second power (a), the following scenarios are possible:
- when the coefficient on the second power is positive (a> 0) - the parabola's arms are directed upwards,
- when the coefficient on the second power is negative (a < 0) - the parabola arms are directed downwards,
- in the case when the coefficient on the second power is equal to zero (a = 0) - the quadratic function reduces to linear function.
A square function can have one, two, or have no zero points. To check the number of zero places (sometimes also called roots), we can calculate the discriminant of a quadratic function (colloquially called delta):

$\Delta=b^{2}-4~a \cdot c$
where:
- $\Delta$ - dicriminant of the quadratic function,
- $a$ , $b$ , $c$ - quadratic function coefficients (numbers just before x², x and free parameter).
then the following scenarios are possible:
- discriminant is negative (Δ <0) - the function has no roots, the graph of the function is a parabola, which is located entirety above the OX axis or under the OX axis,
- discriminant is equal to zero (Δ = 0) - the function has exactly one root, the graph of the function is a parabola whose vertex lies on the OX axis:
  
  $h=\frac{-b}{2~a}$
- discriminant is positive (Δ> 0) - the function has two different roots, the function graph is a parabola, whose arms cross the OX axis:
  
  $x_1=\frac{-b-\sqrt{\Delta}}{2~a}$
  
  $x_2=\frac{-b+\sqrt{\Delta}}{2~a}$
A quadratic function is a special case of polynomial function in which the order is 2.

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