Common Core Standards: Math
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Students should understand that an equation and its graph are just two different representations of the same thing. The graph of the line or curve of a two-variable equation shows in visual form all of the solutions (infinite as they may be) to our equation in written form. When two equations are set to equal one another, their solution is the point at which graphically they intersect one another. Depending on the equations (and the alignment of the planets), there might be one solution, or more, or none at all.
Students can arrive at the correct answer(s) through graphing the functions and plotting their intersection points, creating a table of x and f(x) values, and solving for x algebraically when f(x) = g(x). These strategies should be provided to students and practiced with students so that the connection between graphs and equations are solidified. (We wouldn't want them to be liquefied, now would we?)