In grades 9-12, the mathematics curriculum should include the continued study of functions so that all students can--

• model real-world phenomena with a variety of functions;
• represent and analyze relationships using tables, verbal rules, equations, and graphs;
• translate among tabular, symbolic, and graphical representations of functions;
• recognize that a variety of problem situations can be modeled by the same type of function;
• analyze the effects of parameter changes on the graphs of functions;

and so that, in addition, college-intending students can--

• understand operations on, and the general properties and behavior of, classes of functions.

Focus

The concept of function is an important unifying idea in mathematics. Functions, which are special correspondences between the elements of two sets, are common throughout the curriculum. In arithmetic, functions appear as the usual operations on numbers, where a pair of numbers corresponds to a single number, such as the sum of the pair; in algebra, functions are relationships between variables that represent numbers; in geometry, functions relate sets of points to their images under motions such as flips, slides, and turns; and in probability, they relate events to their likelihoods. The function concept also is important because it is a mathematical representation of many input-output situations found in the real world, including those that recently have arisen as

a result of technological advances. An obvious example is the [x] key on a calculator.

Discussion

To establish a strong conceptual foundation before the formal notation and language of functions are presented, students in grades 9-12 should continue the informal investigation of functions that they started in grades 5-8. Later, concepts such as domain and range can be formalized and f(x) notation can be introduced, but care should be taken to treat these as natural extensions of the initial informal experiences. A function can be described by a written statement, by an algebraic formula, as a table of input-output values, or by a graph. Students need to work with each of these representations and to translate among them.

Since functional relationships are encountered so frequently, the study of functions should begin with a sampling of those that exist in the students' world. Students should have the opportunity to appreciate the pervasiveness of functions through activities such as describing real-world relationships that can be depicted by graphs, reading and interpreting graphs, and sketching graphs of data in which the value of one variable depends on the value of another. For example, students could be provided the graph in figure 6.1 and asked to determine what real-world phenomenon might be represented by the graph (e.g., temperature of an oven). Interpretive activities might include asking them to give an approximate value for A and to indicate what it represents; to determine what event is occurring during the time interval from B to C; or to explain why the graph oscillates for times greater than C.

Fig. 6.1. Oven temperature as a function of time

As a second example, consider the following student-prepared graph (fig. 6.2) and the richness of the setting for encouraging mathematical reasoning and communication. Students could be asked to explain a possible meaning of the vertical intercept (e.g., the cost of the band) or the break in the graph (e.g., the cost of a security officer required for more than 150 students), to determine how many tickets were sold before breaking even, to find the cost of the tickets sold prior to the dance and at the door, or to explain how the graph should be modified if it were discovered that the student treasurer failed to take into account the $50 cost of chaperones.

Fig. 6.2. Dance profit as a function of tickets sold

Students are frequently given experiences graphing functions expressed in symbolic form. It is equally important, however, that they be given opportunities to translate from a graphical representation of a function to a symbolic form. For example, an analysis of the graph in figure 6.2 would lead to the following piecewise definition of the function:

P(t) = 2t - 200 for 0  t < 150      



     = 3t - 400 for 150  t  200

The power of functions to simplify complex situations and to predict outcomes can be demonstrated by observing a phenomenon involving an underlying functional relationship between two variables, gathering and plotting observational data, fitting a graph to the plotted points, using the graph to formulate the relationship between the variables, and then predicting outcomes for unobserved values of one of the variables. For example, students could record the number of swings during a given time period for pendulums of differing lengths, graph the relationship between the number of swings and the length of the pendulum, formulate this relationship, use it to predict the number of swings for pendulums of other lengths, and validate their predictions.

Computing technology provides tools, especially spreadsheets and graphing utilities, that make the study of function concepts and their applications accessible to all students in grades 9-12. This technology makes it possible for students to observe the behavior of many types of functions, including direct and inverse variation, general polynomial, radical, step, exponential, logarithmic, and sinusoidal. All students should use a graphing utility to investigate how the graph of y = af(bx + c) + d is related to the graph of y = f(x) for various changes of the parameters a, b, c, and d. The effects of these changes can best be expressed by geometric transformations, thereby providing another connection between algebra and geometry. This is illustrated for the function f(x) = x ² by the sequence of graphs in figure 6.3.

Fig. 6.3. Altering functions, transforming graphs

College-intending students should develop an understanding of polynomial, rational, algebraic, and transcendental functions, and of those defined piecewise in terms of any of the above. Each of these types of functions should be used to model several problem situations so students can abstract the differences and commonalities in problem situations that are modeled by a given type of function.

Computing technology has made recursively defined functions increasingly important. As a result, college-intending students should develop some facility with functions defined in this manner. For example, n! can be viewed recursively as a function f defined on the nonnegative integers as follows:

f(0) = 0! = 1



f(n) = n! = nf(n - 1) = n(n - 1)! for n  1

Once they have acquired an understanding of various functions, college-intending students should apply techniques to fit curves, that is, functions, to data collected experimentally or supplied in tabular form. If the data are not linear, students should use finite-difference methods, statistical computer packages, and log-log or semi-log transformations to formulate and subsequently make predictions from the function of best fit.

College-intending students also should learn how to combine functions by addition and by composition, and properties of these operations should be analyzed and interpreted graphically. The concept of inverse function should be explored informally by all students as a process of undoing the effect of applying a given function. Furthermore, all students could engage in activities in which they discover and apply the fact that the reflection image of the graph of a one-to-one function across the line y = x is the graph of its inverse. The formal notation and the precise definition of inverse function should be reserved for college-intending students.

Finally, college-intending students should use graphing utilities to investigate informally the surfaces generated by functions of two variables. Such investigations not only contribute to further development of important visualization skills but also foreshadow more advanced work with functions.