Algebra 1 PARCC question: linear functions

Correct answers: B and D.

PARCC evidence statement(s) tested: F-IF.1, subclaim A:

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

The evidence statement references Math Practice 2 in the Common Core: Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

The question tests students’ understanding of the eighth-grade Common Core math standard 8.F.A.1, which states that they should “understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.” It also touches 8.F.B.5, which says they should be able to “describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.”

Although the question begins to tap into HSF.LE.A.1, a high school math standard in “Linear, Quadratic, and Exponential Models,” most of the standards assessed by this question are in eighth grade within the Common Core and, I suspect, most states’ math standards.

The first option, a set of ordered pairs, is incorrect. Two of the ordered pairs have an x value of 3 but have different y values. As stated above, each element of the domain in a linear function must map to exactly one element of the range. If the same x value gives two different y values, the set of ordered pairs does not represent a linear function (or any type of function).

The graph is correct. As long as any line plotted on a set of coordinate axes isn’t straight up-and-down vertical, the graph represents a function. The fact that the graph is a “line” means it’s “linear.” Therefore, this is a perfectly good way to represent a linear function.

The table is incorrect. As with the set of ordered pairs, the same x value yields more than one y value. This table, therefore, doesn’t represent a function, linear or otherwise.

The description of how to find the perimeter of a square is correct. The word problem translates to an equation, which can be plotted as a non-vertical line and therefore represents a linear function, since it meets all the requirements for a function—each valid input has exactly one output—and for being linear. If s = 4, for example, then the perimeter is 16. It can’t be anything but 16, so only one output results from the input of 4. Expressing that in the format y = mx + b would give us something like p = 4s + 0.

The absolute value function is incorrect. It’s not linear, because it looks like a “V” (see below), but it is a function. For each input, whether it’s a positive or negative number, there is exactly one output. On the graph of y = |x|, no points are directly on top of another point, although some points are side-to-side.