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).

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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).

Slope of the line

NameFormulaLegend
Slope from two points lying on the lineShow sourcea=y1y0x1x0 a=\frac{ y_1- y_0}{ x_1- x_0}
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Slope from one point lying on the line (free parameter is needed)Show sourcea=y0bx0 a=\frac{ y_0- b}{ x_0}
  • 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),
  • x0x_0, y0y_0 - point coordinates.

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.

Some facts

  • The linear function is a function that can be presented in the following form:
    y=a x+b y= a~ x+ b
    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 - 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=ba 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 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=0y = 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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