
Powers of x: positive integer powers
On this page, we will review some properties of the functions which are
powers of the independent variable
Many of these functions
will probably be quite familiar, but there are several subtle points
that are worth remembering early on in a "Calculus career". In fact,
some of the features of these very simple building blocks will come
in handy many times in our later discussions of properties of
functions. We will begin the discussion by examining the
simplest of these functions, those of the form
where n is a positive integer.
Below, we illustrate the graph of the function
where
is a positive integer and
is a
constant. By moving the sliders, you can vary the values of
and
Questions:
At this point, you should explore the properties of the function
by seeing what happens when the values of
and
are changed.
Then, answer the questions below. Your answers should be based
exclusively on your observations here. Calculus, derivatives, and
any other tools should not be aplied at this point since we want
to help you develop your "powers of observation" (to use a pun)
of information on a graph. This type of common sense (rather than
mere calculations) will be very useful in many situations.
 Effect of the constant: Set n=2. Increase the
parameter a. Explain what happens to the
graph of the function. Is your observation true also for n=3 ?
for bigger n values ? Explain in words what is the effect of the
constant a on these graphs.
 Effect of the power: Now that we have
explored the effect of the multiplicative
constant, a, we will set it equal to 1 and investigate the effect
of raising the power. Set a=1.
 What is the shape of the graph when n is an
even integer?
 What is the shape when n is odd?
 Can you summarize these observations in terms
of symmetry of the graphs?
 Rates of growth: How can you explain the
following statements in terms of the shape of the graphs?
 From the point
, the function
grows more rapidly than
.
 From the point
, the function
grows more rapidly than
.
Generalize to other powers. How does the rate of growth at
and
depend on
?
 The symmetry of the graphs of
the power functions
lends an important terminology to functions in general.
A function
is called even or
odd if it shares the symmetry properties of the
above odd and even power functions. Now that you know this
little fact, match the description "odd" and "even" with the
following properties of a function.

is ____________ if

is ____________ if
Find other examples of even and odd functions. Not every function
is either even or odd. (But it is an interesting fact that every
function can be expressed as a sum of an even and an odd function.)
 Intersections Which point(s) do all
the graphs of
have in
common
 for even values of n ?
 for odd values of n ?
 for all values of n ?
 Slope at the origin What is the slope
of the graph of
at the origin?
Is this slope the same or different for even and odd values of the
power n?
 Changing slope
Suppose you were an ant walking across the graph of the
function
, starting at x=1 and proceeding to x=1.
Would you be walking uphill ? downhill ? Does it depend on the
value of n? Translate this observation to a comment about how
the slope of the function changes with increasing x values.
Is this different for even and odd values of n?
 Minima
Which of members of the above family of functions
have a minimum value? Which do not?

