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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Some facts#

  • The linear function is a function that can be presented in the following form:
    y=ax+by=a \cdot 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=bax=\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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