Positive Power Functions
Positive Power Functions
In today’s lecture, we continue our theme of studying polynomials and the terms which comprise polynomials.
A positive power function is a function of the form f(x) = xn, where nis a natural number. You should
become familiar with seeing notation like this, where the exponent is not given explicitly, but instead is
represented by a letter. Here, we mean that we are studying the function f(x) = x, x2, x3, x4, ... as a family.
Suppose we were to graph this family of functions all on the same graph. Specifically, let us graph x,x2,
x3, and x4using the following numerical tables as a guide (we leave out the table for the function x):
x x2
-1.5 2.25
-1 1
-0.5 0.25
0 0
0.5 0.25
1 1
1.5 2.25
x x3
-1.5 -3.375
-1 -1
-0.5 -0.125
0 0
0.5 0.125
1 1
1.5 3.375
x x4
-1.5 5.0625
-1 1
-0.5 0.0625
0 0
0.5 0.0625
1 1
1.5 5.0625
When we sketch out these graphs, we see the following traits of the graphs of positive power functions:
•The graphs of positive power functions come in two basic shapes. If nis even, then the graph of xnhas
an axis of symmetry, and that is the vertical line x= 0, the y-axis. If nis odd, then xnhas a point of
symmetry, which is the origin. This means that if the point (x, y) is in the graph of xn, then the point
(−x, −y) is also in the graph. We will discuss these two types of symmetries more later in the lecture.
•For all values of n,xnequals 0 when x= 0, and xnequals 1 when x= 1. So the graphs of all positive
power functions intersect at the origin and at the point (1,1).
•When xis between 0 and 1, the larger nis, the smaller xnis. You can observe this phenomenon but
studying the values of x,x2,x3,x4for x= 0.5 in the tables above. This has the effect that, if mand
nare two natural numbers and m < n, then between x= 0 and x= 1, the graph of xmis above the
graph of xn. We also see that, as ngets larger, the graph of xnlooks more and more like it is following
the x-axis and the vertical line x= 1, so that it almost looks like it has a corner in it.
•When xis greater than 1, the larger nis, the larger xnis. Think of the powers of any natural number
greater than 1 to see this. This means that, if mand nare natural numbers and m<n, then when
x > 1, the graph of xmis below that of xn. We also observe that, the larger nis, the more the graph
of xnfor x > 1 looks like the vertical line x= 1.
Derivatives of Positive Power Functions
Now let f(x) = xn, where nis some natural number. The derivative of f(x) at x=pis
df
dx(p) = npn−1.
In particular:
•If f(x) = x, then df
dx(p) = 1. This should make sense: in this case, f(x) is a linear function. Recall
that the derivative of a function at a point is the slope of the tangent line to the graph of that function
at that point. The graph of a linear function is a line, and, when you think about it, the tangent line
to a line is itself. So the derivative of a linear function is its own slope. In this case, that slope is 1.
•If f(x) = x2, then the formula above tells us that df
dx(p) = 2p. This is consistent with what we already
know, because here f(x) is a quadratic function, with a= 1 and b= 0, and applying the formula for
the derivative of a quadratic function, we get the same result as above.
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