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

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:
$y= a~{ x}^{2}+ b~ x+ c$
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):
$\Delta={ b}^{2}-4~ a~ c$
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.