Polynomial Functions: Concepts & Procedures
Examples of Polynomial
Expressions. Some algebraic
expressions are polynomials, like: 6 x – 2, 5 x^{2}– 17, –41 x^{3} + 7, 2^{1/2} x^{7}– 4, 3 x(x^{2} – π), and
the number –12. Polynomial
jargon can be confusing.
Examples of Nonpolynomial
Expressions. Some algebraic expressions are not polynomials,
like: x^{1/2}, 3 x^{0.2} – 17, –x^{–1} + 7, x^{2} – x
^{–2}– 4, and x.
Degree. The degree of
a polynomial is the highest power of x that appears in the
polynomial. When the degree of a polynomial
is an even number, the polynomial is said to be of even degree.
When the degree of a polynomial is an odd number, the polynomial
is said to be of odd degree.
Power
Functions. Any function of the form f (x)=k x^{p}, where k and p are
realvalued constants,
is called a power function. The constant k is called
the coefficient of the power function. All power functions with k=1
go through the origin, (0,0), and the point (1,1); in addition,
all even degree power functions go through the point (–1,1)
while all odd degree power functions go through the point (–1,–1).
Polynomials. Polynomial
functions are sums of power functions with nonnegative integer
powers. Note that a function given in factored form, like f (x)=(x^{
3} – 1)^{2}, is
considered a polynomial because it can be expanded to a sum of
power functions with nonnegative integer powers: f (x)=
x^{6}– 2 x^{3} + 1.
Leading
Term. In a polynomial,
the power term with the highest power of x is called the leading
term.
Leading
Coefficient. In a polynomial,
the coefficient of the power term with the highest power of x is
called the leading coefficient.
Intercepts. Any
point where a graph intersects a coordinate axis is called an intercept of
the graph.
xIntercepts. Any
point where a graph intersects the xaxis is called an xintercept of
the graph. Aside from the graph of the constant function f (x)=0,
the maximum
number of xintercepts is
the degree of the polynomial while the minimum
number of xintercepts is
none if the degree is even and one if the degree is odd. Nonpolynomials
may not have any.
yIntercepts. Any
point where a graph intersects the yaxis is called a yintercept of
the graph. Polynomial functions always have exactly one yintercept.
Nonpolynomials may not have any.
Complete
Graph. A complete graph is one that
includes all the important mathematical features. For polynomial
functions, this means the viewing window should be just large
enough to display all x and yintercepts, all
maxima and minima, and all points of inflections. When you enter
a polynomial function and click the GraphIt button, WebGraphing.com automatically
displays a complete graph. Armed with this knowledge, you can
select different viewing windows to zoom in on particular
aspects of the graph or refine it more to your liking.
Zoom
In/Out (Resizing the Viewing Window). If you wish to graph
again with a different display window, you can do this in two
ways: (1) you can enter the lower and upper xbounds only
and let WebGraphing.com determine optimal ybounds
(Smart Zooming) that include
all the important mathematical features contained within your xbounds,
or (2) you can enter both the lower and upper x and ybounds
to choose the precise viewing window. The first option is especially
useful if you are zooming in on a peak or valley or a change
in concavity, since
in that case WebGraphing.com chooses an optimal
range of yvalues. The choice of an appropriatelychosen
display can be important to visually verify, beyond the indicated
colorcoding, underlying features like maxima, minima, and points
of inflection. The following example illustrates this with a
fifth degree polynomial.
Local Maxima, Local Minima, and Concavity Predicted by Color
Coding
over the interval (–5,5)
Local Maxima, Local Minima, and Concavity Exposed by
Smart Zooming from Xmin=–4.5 to Xmax=3.4
Ratio
of the height to width unit lengths (or ratio of the width to
height unit lengths). On the display of a graph, the ratio
between the plotted range of yvalues to the plotted range
of xvalues can vary. For example, if the plotted range
of yvalues is from −10 to 10 and the plotted range
of xvalues is from −1 to 1, then the ratio of the
height to width unit lengths is 10 to 1. When the ratio is very
large or very small, there may be some distortion that can help
or possibly hinder the reading and interpretation of a graph.
For example, two perpendicular lines will not "look" perpendicular
if they are plotted with a ratio of 10 to 1 (unless, of course,
they are horizontal and vertical). To alert you to the possibility
of distortion, we report the ratio between the plotted range
of yvalues to the plotted range of xvalues (or
viceversa) when it is greater than 5 to 1.
Domain
and Range
Domain. The domain of
a function is the set of all permissible xvalues. For any
polynomial, the domain consists of all real numbers, (−∞, ∞). It
is worth noting that nonpolynomial functions often have restrictions
on their domains.
Range. The range of
a function is the set of all yvalues attained. The range
of an odd degree polynomial is always all
real yvalues, −∞ < y < ∞, while
the range of an even degree polynomial, excluding
0 degree polynomials, can be described using inequalities either
in the form of −∞ < y ≤ global
maximum, or global minimum ≤ y < ∞. Often,
the range is expressed using interval notation, in which case the
interval needs to be visually understood as a yinterval.
Any polynomial of degree 0 is a constant function, expressible
as f (x)=k, where k is
a constant, in which case the range is the single yvalue, {k}.
End
Behaviors
End
Behaviors. The term end behaviors refers to the behaviors
of a graph at the left and right tails (when x is
large: x→−∞ or x→∞ ).
For a polynomial, the behaviors follow those of its leading
term and, for degrees of 1 or more, there are only four possibilities,
The four possibilities are modeled by the graphs of the four
power functions: y=±x and y=±x
^{2}. If
the degree is odd: (1) when the leading
coefficient is positive, the behaviors follow those of the odd
degree polynomial y=x (↓↑:
down on the left, up on the right), (2) when the leading coefficient
is negative, the behaviors follow those of the odd degree polynomial
y=−x (↑↓:
up on the left, down on the right). If the degree
is even: (1) when the leading coefficient is
positive, the behaviors follow those of the even degree polynomial
y=x^{2} (↑↑:
up on the left, up on the right), (2) when the leading coefficient
is negative, the behaviors follow those of the even degree polynomial
y=−x^{2} (↓↓:
down on the left, down on the right). Any polynomial of degree
0 is a constant function, and its end behavior at each tail is
to remain at its constant value.
Symmetry
Symmetry
With Respect to the yAxis. The graph of y=f (x) is
symmetric with respect to the yaxis if f(–x)=f(x). In
this case, the function is called even.
Symmetry
With Respect to the Origin. The graph of y=f (x) is symmetric
with respect to the origin if f (–x)=–f (x). In
this case, the function is called odd.
Maxima
and Minima
Local Maxima. The yvalue f (c)
is a local maximum value (also called relative maximum
value) of f if there is an open interval containing
the xvalue c where f (c) ≥ f (x). In
this regard, the First Derivative Test states that when
the graph of a polynomial function y=f (x) is increasing on
the immediate left of the number x=c and decreasing on
the immediate right of the number x=c, then
the value of f at c is locally the
largest, i.e., f (c) is
a local maximum.
Local Minima. The yvalue f (c)
is a local minimum value (also called relative minimum
value) of f if there is an open interval containing
the xvalue c where f (c) ≤ f (x). In
this regard, the First Derivative Test states that when
the graph of the polynomial function y=f (x) is decreasing on
the immediate left of the number x=c and increasing on
the immediate right of the number x=c, then
the value of f at c is locally the
smallest, i.e., f (c) is
a local minimum.
Determining Local Maxima and Local Minima. Any
value of x for which f '(x)=0 is
called a zero of f '. For polynomials,
the local maxima and local minima can only occur at the realvalued
zeros of f ' . These zeros separate the real
number line into intervals. To determine all local maxima and local
minima, select a
convenient "Test Value" on each such interval and determine
the sign of f '(Test Value). At each zero
of f ', there are three possibilities:
1. If the sign of f '(Test Value) is
positive (+) on the left
side (interval to the left) of a zero of f ' and
negative (−) on
the right side, suggesting visually that the function is increasing on
the left side and decreasing on
the right side, then f has
a local maximum at that zero.
2. If the sign of f '(Test Value) is
negative (−) on the left side of a zero of f ' and
positive (+) on the right side, suggesting visually
that the function is decreasing on the left side and increasing
on the right side, then f has
a local minimum at that zero.
3. If there is no change in sign of f '(Test
Value) from either side of a zero of f ' to
the other side, then the zero is not a local maximum or local
minimum for f.
In the latter case, if the sign of f '(Test
Value) is (+) on the left side of a
zero of f ' and (+) on the
right side as well, this suggests visually that the function
is increasing on the left side and increasing on the right
side. So, the curve has a horizontal tangent at that zero but
the point of tangency is not a turning point.
Alternatively, if the sign of f '(Test
Value) is (−) on the left side
of a zero of f ' and (−) on
the right side as well, this suggests visually that the function
is decreasing on the left side and decreasing on the right
side. Once again, the curve has a horizontal tangent at the
zero but the point of tangency is not a turning point. In
both of these cases, the horizontal tangent cuts through
to the other side of the curve. Visually, this suggests a
change in concavity, which is treated using the second derivative.
Global Maxima. A
local maximum yvalue f (c)
is the global maximum value (also called the absolute
maximum value) if f (c) ≥ f (x) for
all x in the domain of f.
Global Minima. A
local minimum yvalue f (c)
is the global minimum value (also called the absolute
minimum value) if f (c) ≤ f (x) for
all x in the domain of f.
Increasing/Decreasing Intervals
Increasing. Geometrically
speaking, a function y=f (x) is increasing on
an interval if the function is rising over the interval as you
look at the graph from left to right. By definition, y=f (x) is increasing on
an interval I if f (a) < f (b) whenever a < b, for a, b in I. This
is true if f ′(x) > 0 at
interior points of the interval I. The latter condition
is adequate (sufficient) to establish that a function is increasing,
but it does not always work to verify the "increasing" property,
as indicated by the example of the cubic f (x)=x^{3}, where f '(0)=0 but y=f (x) is
increasing on any interval containing 0. The increasing segments (portions
or arcs of a graph) of a graph are shown on Webgraphing.com in red.
Decreasing. Geometrically
speaking, a function y=f(x) is decreasing on
an interval if the function is falling over the interval as you
look at the graph from left to right. By definition, y=f (x) is decreasing on
an interval I if f (a) > f (b) whenever a < b, for a, b in I. This
is true if f ′(x) < 0 at
interior points of the interval I. The latter condition
is adequate (sufficient) to establish that a function is increasing,
but it does not always work to verify the "decreasing" property,
as indicated by the example of the cubic f (x)=– x^{3}, where f '(0)=0 but y=f (x) is
decreasing on any interval containing 0. The decreasing segments
of a graph are shown on Webgraphing.com in blue.
Determining Increasing
and Decreasing Intervals. To determine the intervals
of increase and decrease for a polynomial function y=f (x),
first find all the realvalued zeros of f '. These
zeros separate the real number line into intervals. Select a convenient Test
Value on each such interval and determine the sign of f '
(Test Value). If the sign is positive (+), then
the function is increasing on interval. If the sign is negative (−), then
the function is decreasing on the interval. Based on these considerations,
the intervals of increase and decrease can be determined by creating
a table with zeros of f 'and test values.
Points of Inflection. A
point on a graph where (1) there is a change in concavity and (2)
the graph has a tangent is called a point of inflection.
Since polynomials always have tangents at all their points, this
latter condition has importance only for nonpolynomials.
Determining Points of Inflection. Any
value of x for which f ''(x)=0 is
called a zero of f ''. For polynomials,
the points of inflection can only occur at realvalued zeros of f ''. These
zeros separate the real number line into intervals. To determine
all points of inflection, select
a convenient "Test Value" on each such interval and determine
the sign of f ''(Test Value). At each
zero of f '', there are three possibilities:
1. If the sign of f ''(Test Value) is
positive (+) on the left side (interval to the left) of a zero
of f '' and negative (−) on
the right side (interval to the right), indicating visually that
the function is concave
up on the left side and concave
down on the right side, then f has
a point of inflection at that zero.
2. If the sign of f ''(Test Value) is
negative (−) on the left side of a zero of f '' and
positive (+) on the right side, indicating visually
that the function is concave down on the left side and concave
up on the right side, then f has
a point of inflection at that zero.
3. If there is no change in sign of f ''(Test Value)
from one side of a zero of f '' to the other
side, then the zero is not a point of inflection for f.
Concave
Up/Concave Down Intervals
Concave Up. Geometrically
speaking, an arc of a graph is concave up if every line
segment connecting two points on the arc lies above the arc.
From the perspective of calculus, this happens when the tangent
lines to y=f (x) are
locally below the arc, that is, when the first derivative, f '
, is increasing. In turn, the first derivative is increasing
when its derivative, the second derivative of f, is positive: f ′′(x) > 0.
Thus, consistent with the underlying geometry, we say that a
function is concave up on an interval if its second derivative
is positive at the interior points of that interval. The concave
up arcs of a graph are shown on Webgraphing.com in purple.
Concave Down. Geometrically
speaking, an arc of a graph is concave down if every line
segment connecting two points on the arc lies below the arc.
From the perspective of calculus, this happens when the tangent
lines to y=f >(x) are
locally above the arc, that is, when the first derivative, f '
, is decreasing. In turn, the first derivative is decreasing
when its derivative, the second derivative of f, is negative: f ''(x) < 0.
Thus, consistent with the underlying geometry, we say that a
function is concave down on an interval if its second
derivative is negative at the interior points of that interval.
The concave down arcs of a graph are shown on Webgraphing.com in green.
Determining Intervals
of Concavity. For a polynomial function y=f (x), to
determine the intervals where the graph is concave up and
concave down, first find all the realvalued zeros of the
second derivative, f ''. These zeros
separate the real number line into intervals. Select a
convenient Test Value on each such interval and
determine the sign of f ''(Test Value). If
the sign is positive (+), then the function is concave
up on the interval. If the sign is negative (−), then
the function is concave down on the interval. Based
on these considerations, the intervals where the function
is concave up and concave down can be determined by creating
a table with zeros of f '' and test values.
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United States Patent Numbers 7,432,926, 7,595,801, & 7,889,199.
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