Algebra test items - Graphs of rational functions

TIMSS (Trends in international Math and Science Study) classifies test items in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

In my earlier post about this topic on using the TIMSS Assessment Framework for constructing test items I presented a set of questions about zeros of cubic polynomial function. Here are three more test items about graphs of rational function based on the framework. Note that questions should be independent of each other, that is, an answer in one item should not serve as clue to the other items. I only used the same rational function here to highlight the differences among the cognitive domains – knowing, applying, reasoning.

Knowing

What may be the equation of the graph below?

Applying

The graph above the x-axis is function f and the graph below the x-axis is function g. Which of the following equations describes the relationships between f and g?

a. g(x) = f(-x) b. g(x) = f^-1(x) c. g(x) = f^-1(-x) d. g(x) = -f(x) e. g(x) = /f(x)/

Reasoning

Carlo drew the figure below by graphing two functions on the same coordinate axes. The graph on the left is f(x) = 4/x². Which of the following function is represented by the other graph on the right (the blue one)?

a. $g(x)=\frac {4}{x^2}$ b. $g(x)=4+\frac {4}{x^2}$ c. $g(x)=\frac {4}{(x-2)^2}$ d. $g(x)=\frac {4}{(x-4)^2}$ e. $g(x)=\frac {4}{(x+4)^2}$