Functions
The next major topic in algebra is the concept of a function, its definition and
the
detailed study of linear and quadratic functions . Students usually have some
trouble
coming to grips with this concept. Other than the need of many examples to
illustrate its
many-sided ramifications, students should also be shown why functions are
indispensable.
One can ask how to describe the temperature of a cup of freshly brewed coffee in
its first
15 minutes, for example, or the position of a piece of chalk thrown across the
classroom.
Much of mathematics grows out of necessity, and students should get to know this
fact.
Two common practices in the teaching of functions should be avoided. One is to
say
that a function can be re presented by (pictures of) graphs, tables, rules and
formulas.
Since in school algebra the functions are almost always real-valued functions
defined
on the number line, tables and the pictures of graphs can suggest certain
aspects of a
function, but can never give a complete representation of a function. A second
one is
to over-emphasize the importance of the concept of a relation, especially the
difference
between a relation and a function. Some books even go so far as to define the
domain and
range of a relation. From the point of view of what mathematics needs, the
concept of a
relation deserves to be mentioned for students’ benefit, but it is far from a
main topic in
school algebra. We repeat what was said earlier about
factoring quadratic polynomials:
if a little bit is good, it does not fol low that a lot is better .
It may seem repetitious, but it needs to be said once more that one must give an
explicit
definition of the graph of a function and use it to prove theorems. For a
real-valued
function of one variable, its graph is the subset of the plane consisting of all
ordered pairs
{(x, f(x))} where x is a member of the domain of f. Emphasizing the definition
of a
graph addresses a frequently-asked-question by students: if the graph of f
crosses the
x-axis at (m, 0), why is m a solution of f(x) = 0? This is because (m, 0), being
a point
on the graph of f, must be of the form (t, f(t)), by the definition of the
graph. Therefore
(m, 0) is equal to (m, f(m)), so that 0 = f(m), and m is a solution of f(x) = 0.
The graphs of linear functions, that is, those of the form f(x) = cx + k (c, k
being constants), are lines. This follows from what we know about the graphs of
linear
equations in two variables , because the graph of a linear function f(x) = ax + k
is the
same as the graph of the linear equation of two variables y = ax + k. A special
class
of linear functions, those without constant term (i.e., k), are especially
important in
middle school mathematics. They underlie all conside rations of constant rate:
constant
speed, for example, is the statement that there is a constant v, so that if the
distance
traveled from the origin at time 0 to time t is f(t), then f(t) = vt. It also
underlies all
the problems connected with proportional reasoning, but since this topic is not
well
understood in mathematics education at the moment, we pause to give a brief
discussion.
The general understanding of proportional reasoning (cf. [NRC2001], p. 241–244),
to
the effect that it is about “understanding the underlying relationships in a
proportional
situation and working with these relationships” (loc. cit., p. 241), appears to
be related
to some misconception about the formulation of mathematical problems. To explain
this misconception, consider a prototypical proportional-reasoning problem such
as the
following.
A group of 8 people are going camping for three days and need to carry their
own water. They read in a guide book that 12.5 liters are needed for a party
of 5 persons for 1 day. How much water should they carry?
On one level, proportional reasoning in the way it is usually understood is
about the
ability to infer from the given data that, the amount of water con sumed by n
individuals
per day is proportional to n. In symbolic terms , if f(n)
is the amount of water consumed
by n in dividuals per day (where n is a positive integer), then a student who is
capable of
proportional reasoning would supposedly conclude that
for all positive integers m and n. This is an unreasonable
expectation, because there is
no logical reasoning to justify the leap from the given data of the problem to
the above
proportional relationship. To understand what this proportional relationship
means, let
k be the common value of all these quotients. Then we have f(n) = kn no matter
what
the integer n is. Thus proportional reasoning would include the recognition that
this
function f(n), as a function defined on the positive integers, is a linear
function. Since it
is customary to regard this function as defined on the real numbers , we have a
linear
function f(x) = kx for all real numbers x. Furthermore, letting x be 1, then we
have
f(1) = k. This is the statement that each person drinks k liters of water per
day.
It is important to emphasize this fact, namely, that a student well versed in
proportional
reasoning is assumed to be one who can infer from the given data that every
person
in the camping trip drinks the same amount of water per day, namely, k liters.
In other
words, we believe that students with conceptual understanding of proportional
reasoning
would be able to draw such a far-fetched conclusion. Indeed, even young kids can
see that some people drink lots of water and others very little. This is not the
kind of
mathematical reasoning we want to postulate as desirable for students to
acquire.
Let us consider the possibility that such faulty reasoning is solely the result
of the faulty
formulation of the problem. How would one formulate a problem having at least
some
contact with the “real world” along this line? A more responsible, and
mathematically
more accurate, formulation of the problem might read something like this :
A group of 8 people are going camping for three days and need to carry their
own water. They read in a guide book that 12.5 liters are needed for a party
of 5 persons for 1 day. If one infers from the guide book that these figures
provide a rough estimate of the amount of water consumed by a party of any
size on any day, roughly how much water should they carry?
It goes without saying that the words, “provide a rough estimate”, are nothing
but code
words for which students need precise and explicit explanations. One expects,
therefore,
that students would receive instruction on reasoning of the following kind. The
key words
in the problem are those describing the amount of water
consumed by a party of “any
size on any day”. These are words that convey the generality of the message of
the guide
book (see the discussion of generality at the beginning of this article). If any
5 persons
drink 12.5 liters per day, then students need to be made aware of the commonly
accepted
interpretation of this statement to mean that any person drinks roughly
liters
on any day. Hence, for any positive integer n, n persons would drink, again
roughly,
n × 2.5 = 2.5n liters on a given day. Therefore it makes sense to define a
function f so
that f(n) is roughly the amount of water n persons consume on a given day, and
we saw
that we had an expression for this f: f(n) = 2.5n liters.
While the preceding discussion succeeds in making more sense of the conclusion
that
each person drinks 2.5 liters per day, the more important message is the need to
make
explicit, in one way or another, the underlying linear relationship in problems
related to
proportional reasoning. One must set some ground rules so that students are not
required
to guess the linear relationship underlying the problem but that the linear
relationship is
made clear in some fashion. We do not wish to enforce the rigid requirement that
a linear
function be handed to students in each problem of this type; such a requirement
would be
anti-educational. Rather, there should be universal recognition that linear
relationships
cannot be taken for granted, and students need explicit instructions as to when
they take
place.
Remember: mathematics has to be precise. Some may feel that when mathematics
is taught with realistic contexts, students will build their own ideas and make
sense of
problems mathematically and make use of them to solve difficult problems.
Perhaps,
and perhaps not. But mathematics is not about saying things that would most
likely
be understood. Rather, it is about saying things in a way that would be
completely and
unequivocally understood. For this reason, if we want students to know that the
“underlying
relationship in a proportional situation” is a linear function, we must ensure
that
it is clearly understood as such. A correctly formulated mathematical problem is
explicit
about its assumptions, because being explicit about assumptions is a basic
requirement
of mathematics.
Once this point is understood, a classroom discussion of word problems related
to
proportional reasoning, whether correctly formulated or not, from the point of
view of
linear functions should be both revealing and rewarding.
From linear functions we go to quadratic ones. The graph of a linear function is
a line,
but what is the graph of a quadratic function? From our prior experience with
quadratic
equations, we first look at a special class of quadratic
functions. The simplest is f(x) = x2,
and the starting point of our discussion is that the graph of f(x) = x2 is
known. The next
is f(x) = ax2 (for a real number a), then f(x) = ax2+q, and finally f(x) =
a(x+p)2+q,
for fixed constants a, p, and q. Let us concentrate on the last. From f(−p+s) =
f(−p−s)
for any real number s, we see that the graph of f is symmetric with respect to
the vertical
line x = −p, and that the graph has its lowest (resp., highest) point at (−p,
f(−p)), if
a > 0 (resp., a < 0). Of course f(−p) = q. Moreover, the translation of the
plane given
by (x, y) → (x+p, y −q) carries the graph of f onto the graph of the simple
quadratic
function 
, so there is no doubt about the shape of the graph of f. So
at least
for simple quadratic functions expressible as f(x) = a(x+p)2+q, the graph is
completely
understood, and therewith, the function itself is completely understood. In
fact, we can
trivially read off the two points 
and 
on the x-axis at which the function
f is equal
to 0:
These are of course the roots of the quadratic equation
a(x+p)2+q = 0. (See the previous
discussion on the relationship between the point where the graph of f crosses
the x-axis
and the solution of f(x) = 0.)
The fundamental theorem about quadratic functions is that, by the technique of
completing
the square, every quadratic function f(x) = ax2 + bx + c can be written in the
form f(x) = a(x + p)2 + q. In fact, simply let
and
It therefore follows from a comment in the last paragraph
that the graph of the function
f(x) = ax2 + bx + c can be made to coincide with the graph of
by the
use of a translation. Moreover, we also know from the last paragraph that the
graph of
f(x) = ax2+bx+c has an axis of reflection symmetry along the vertical
line 
,
and its lowest point (respectively, highest point) if a is positive
(respectively, negative) is
Thus by use of completing the square, one achieves a more
conceptual way to look at the
graph of f as regards shape and location.
Using the above expression of the zeros and
in terms of p and q, we get immediately
that 
for
