In a linear equation, each term is either a constant or the product of a constant and a variable - this means that, as x increases by a fixed amount, y also increases by a fixed amount. Let's take y = 5x as an example. "5x" is the product of a constant (i.e. 5) and a variable (i.e. x). Let's see what happens to y as x increases by 1, starting from 0. At x = 0, y also equals 0. At x = 1, y = 5 (that is, y increased by 5). When x = 2, y = 10. That is, y increased by 5 again. Each time we add 1 to x, y increases by the same amount. That is a property of linear functions.
Now let's look at your example: y = -2x + x^2. Right off the bat, we know that x^2 is not a constant or a simple product - it's an exponent. This means that your y value will not increase by a fixed amount. When x = 0, y also equals 0. When x = 1, y = -1. That is, y changed by -1. Now if x = 2, y = 0 again - thus, change of +1. And if x = 3, y = 3 (y changed by 3). That fact that y changes by a different amount for every unit increase in x means that this is not a linear equation. And the non-linear part of the equation is the "x^2". Had it just been, y = -2x, then it would have been linear.
Robert A.
09/10/15