Mathematical tables: quadratic function formulas
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

NameFormulaLegend
Exponential function in general formShow sourcey=ax y={ a}^{ x}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Exponential function with base e (often written as exp(x))Show sourceexp(x)=ex exp(x)={ e}^{ x}
  • exp(x)exp(x) - value of exponent function,
  • xx - function argument (called also independent value),
  • ee - number e (mathematical constant, base of natural logarithm).
Homographic function in general formShow sourcey=a x+bc x+d y=\frac{ a~ x+ b}{ c~ x+ d}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0).
Function b/xShow sourcey=bx y=\frac{ b}{ x}
  • yy - 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),
  • xx - function argument (called also independent value),
  • b - coefficient b.
Linear function in slope-intercept formShow sourcey=a x+b y= a~ x+ b
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb - linear function coefficients (slope and free parameter).
Linear function in point-slope formShow sourcey=a(xx0)+y0 y=\mathrm{a}\left( x- x_0\right)+ y_0
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • x0x_0, y0y_0 - point coordinates.
Linear function in constant-slope formShow sourceyy0xx0=y1y0x1x0\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Zero of the linear function from constant-slope formShow sourcex=y0(x1x0)y1y0+x0 x=\frac{ y_0\cdot\left( x_1- x_0\right)}{ y_1- y_0}+ x_0
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Quadratic function in standard formShow sourcey=a x2+b x+c y= a~{ x}^{2}+ b~ x+ c
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).
Quadratic function in factored formShow sourcey=a(xx1) (xx2) y=\mathrm{a}\left( x- x_1\right)~\left( x- x_2\right)
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • x1x_1, x2x_2 - function zero points (arguments, for which function has value of zero, solutions of the f(x)=0 equation).
Quadratic function in vertex formShow sourcey=a(xh)2+k y={\mathrm{a}\left( x- h\right)}^{2}+ k
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • hh, kk - coordinates of the parabola vertex (at this point function reaches its local extremum).

Function discriminant

NameFormulaLegend
Discriminant of homographic functionShow sourceD=adbc D= a\cdot d- b\cdot c
  • DD - 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 functionShow sourceΔ=b24 a c \Delta={ b}^{2}-4~ a~ c
  • Δ\Delta - dicriminant of the quadratic function,
  • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).

Zeroes of the function (roots)

NameFormulaLegend
Zero point of homographic functionShow sourcex=ba x=\frac{- b}{ a}
  • xx - 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 functionShow sourcex=ba x=\frac{- b}{ a}
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • bb - free parameter (linear function crosses the OY axis at (0,b) point),
  • xx - 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 formShow sourcex=x0y0a x= x_0-\frac{ y_0}{ a}
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • x0x_0, y0y_0 - point coordinates.
Zero of the linear function from constant-slope formShow sourcex=y0(x1x0)y1y0+x0 x=\frac{ y_0\cdot\left( x_1- x_0\right)}{ y_1- y_0}+ x_0
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
The first root of the quadratic functionShow sourcex1=bΔ2 a x_1=\frac{- b-\sqrt{ \Delta}}{2~ a}
  • x1x_1 - the first root of the function,
  • bb - coefficient before power of one (number just before x),
  • aa - coefficient before power of two (number just before x2),
  • Δ\Delta - dicriminant of the quadratic function.
The second root of the quadratic functionShow sourcex2=b+Δ2 a x_2=\frac{- b+\sqrt{ \Delta}}{2~ a}
  • x2x_2 - the second root of the function,
  • bb - coefficient before power of one (number just before x),
  • aa - coefficient before power of two (number just before x2),
  • Δ\Delta - dicriminant of the quadratic function.

Parabola vertex

NameFormulaLegend
The x coordinate of parabola vertexShow sourceh=b2 a h=\frac{- b}{2~ a}
  • hh - x coordinate of the parabola vertex (for this argument the function reaches its local extremum),
  • bb - coefficient before power of one (number just before x),
  • aa - coefficient before power of two (number just before x2).
The y coordinate of parabola vertexShow sourcek=Δ4 a k=\frac{- \Delta}{4~ a}
  • kk - y coordinate of the parabola vertex,
  • Δ\Delta - dicriminant of the quadratic function,
  • aa - 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 x2+b x+c y= a~{ x}^{2}+ b~ x+ c
    where:
    • yy - function value (the function value at single point x, often marked as f(x)),
    • xx - function argument (called also independent value),
    • aa, bb, cc - 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):
    Δ=b24 a c \Delta={ b}^{2}-4~ a~ c
    where:
    • Δ\Delta - dicriminant of the quadratic function,
    • aa, bb, cc - 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=b2 a 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:
      x1=bΔ2 a x_1=\frac{- b-\sqrt{ \Delta}}{2~ a}
      x2=b+Δ2 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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