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How are linear functions identified from their equations?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

A linear function is identified by its algebraic structure, where all variables appear only in the first degree—meaning they are not raised to any power other than one—and are not multiplied together.12

In a linear equation, these variables are multiplied by constants and combined only through addition or subtraction. When expressed as a function of two variables, a linear equation often takes the slope-intercept form:1

$y = m x + b$

In this form:
* m represents the slope of the line.
* b represents the y-intercept.

If an equation contains a variable raised to a power other than one (such as $y^{2}$ ) or involves variables multiplied by each other, it is not considered linear. When graphed on Cartesian coordinates, a linear equation of two variables will always represent a straight line.2