Linear vs. Nonlinear Functions

The simplest way to identify a linear function is to look at its graph. A linear function always forms a single, straight line that continues forever in both directions. This happens because the relationship between the input ($x$) and the output ($y$) never changes speed. A nonlinear function, however, is any function that does not form a straight line. These graphs might be curves, 'U' shapes (parabolas), or even 'V' shapes. If you have to turn your pencil to follow the path, it is nonlinear.

You can spot a linear function just by looking at the exponents in its equation. A linear equation can always be written in the form $y = mx + b$. The most important rule? The variable $x$ must have an exponent of exactly $1$ (though we usually don't write the $1$). If you see $x^2$, $\sqrt{x}$, or $x$ in the denominator (like $\frac{1}{x}$), the function is nonlinear.