9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

In a perfect world, we would have explicit equations for all relationships (and we don't just mean the Facebook official ones). We'd just plug in our effort and calculate our reward. But as you may have realized, life is much less predictable and far from perfect. Even still, that doesn't mean we can't make sense out of it.

Students should be able to compare two functions even when they're both represented differently. To do this successfully, they have to be able to translate between an equation, a graph, a bunch of words, and a table of values, and understand how certain aspects of one representation impact the rest.

Generally, students should start by knowing the difference between polynomial, linear, quadratic, exponential, and rational functions, and be able to identify them by equation and by graph. This means that given a parabolic curve, students should automatically look for equations in the form of y = ax² + bx + c.

More specifically, a function f(x) that has a y-intercept of 4 would need to have an equation such that f(0) = 4. Similarly, a table of values for this function would be expected to have the point (0, 4).

The struggles students might face could be traced back to the different representations of functions. Students may have particular difficulty with one type of representation and as such, may have trouble with conversion. If this becomes a problem, try going over each type of representation side by side, highlighting the corresponding parts of each and matching them like a giant game of connect-the-dots.

Students should also know the accuracy of each representation. For instance, a table of values can't conclusively define a certain type of function, while a graph can't pinpoint intercepts with certainty. An equation is the most accurate and useful when defining a function, and students should make use of that.

If they can transform any graph, table of values, or description into a mathematical equation that describes the function, they should be good to go.