You can represent all algebraic equations graphically on a "coordinate plane"--in other words, by plotting them relative to an x-axis and a y-axis. The "domain," for example, entails all possible values of "x"--the entire possible horizontal extent of the equation when graphed. The "range," then, represents the same idea, only in terms of the vertical y-axis. If these terms confuse you in words, you can also graphically represent them, which makes them much easier to contemplate.
Find a specific equation to examine. Consider the equation "y=x^2 + 5."
Plug the numbers "-10," "0" "6" and "8" into your equation for "x." You should come up with 105, 5, 41 and 69. Plug some different numbers in and see if you notice a pattern.
Consider the definition of "range"--in layman's terms, all possible values of "y" that might occur in an equation. Think about which values of "y" are impossible for this equation, keeping in mind your results. You should determine that for "y=x^2 + 5," "y" must be greater than or equal to 5, no matter the value of "x" you input.
Plot the equation on your graphing calculator for further illustration. Notice that the parabola (the name of the shape this equation forms) bottoms out at 5 (when the "x" value is 0). Observe that values extend infinitely upward on either side of this minimum--it isn't possible that any lower "range" values exist.
Repeat these instructions using the equations: "y=x + 10," "y=x^3 - 20" and "y=3x^2 - 5." Your ranges for the first two equations should be "all real numbers," while the third should be greater than or equal to -5.
- graph image by Roman Sigaev from Fotolia.com