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How to Compare Linear & Nonlinear Equations

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    • 1). Standardize the forms of the equations to assist in the comparison. If, for example, you want to compare "y = -4x + 5" and "y = 3x + 5 + x^2," you should first order the second equation like the first --with more complex terms before simple ones, or "y = x^2 + 3x + 5."

    • 2). Compare the equations based on the extent to which you believe their graphs will rise or fall. The equation "y = -4x + 5," for example, seems like it should fall, since the sign before the most complex term is negative. The other equation, "y = x^2 + 3x + 5," has a positive sign, which indicates it will rise rather than fall.

    • 3). Plug a range of numbers into the equations to see how they vary in different parts of the coordinate plane. A good example might be (-2, -1, 0, 1, 2). For the first equation, do this as follows: y = -4(-2) + 5 = 8 + 5 = 13; y = -4(-1) + 5 = 4 + 5 = 9; y = -4(0) + 5 = 0 + 5 = 5; y = -4(1) + 5 = -1 + 5 =; y = -4(2) + 5 = -8 + 5 = -3. Notice how more negative "x" values produce higher positive "y" values. Plug the range into the second equation: y = (-2)^2 + 3(-2) + 5 = 4 - 6 + 5 = 3; y = (-1)^2 + 3(-1) + 5 = 1 - 3 + 5 = 3; y = (0)^2 + 3(0) + 5 = 0 + 0 + 5 = 5; y = (1)^2 + 3(1) + 5 = 1 + 3 + 5 = 9; y = (2)^2 + 3(2) + 5 = 4 + 6 + 5 = 15. For the second equation, it would appear that the curve starts off level and begins to rise rapidly. Notice that for the value x = 0, the equations are equal, with values of y = 5.

    • 4). Graph the equations to observe the similarities and differences visually. If you don't have a graphing calculator on hand, there are free, online versions available. Enter the equations as follows: -4x + 5; x^2 + 3x + 5. Notice that both equations cross the vertical y axis at the same point and that the curve does not start off flat, but rather dips and starts rising again between the x values of -1 and -2. It shows symmetry, while the flat line does not.

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