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Polynomial Functions: Concepts & Procedures

Examples of Polynomial Expressions. Some algebraic expressions are polynomials, like: 6 x – 2, 5 x2– 17, –41 x3 + 7, 21/2 x7– 4, 3 x(x2 – π), and the number –12. Polynomial jargon can be confusing.

Examples of Nonpolynomial Expressions. Some algebraic expressions are not polynomials, like: x1/2, 3 x0.2 – 17, x–1 + 7, x2 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 xp, where k and p are real-valued 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)= x6– 2 x3 + 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.

x-Intercepts. Any point where a graph intersects the x-axis is called an x-intercept of the graph. Aside from the graph of the constant function f (x)=0, the maximum number of x-intercepts is the degree of the polynomial while the minimum number of x-intercepts is none if the degree is even and one if the degree is odd. Nonpolynomials may not have any.

y-Intercepts. Any point where a graph intersects the y-axis is called a y-intercept of the graph. Polynomial functions always have exactly one y-intercept. 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 y-intercepts, 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 x-bounds only and let WebGraphing.com determine optimal y-bounds (Smart Zooming) that include all the important mathematical features contained within your x-bounds, or (2) you can enter both the lower and upper x- and y-bounds 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 y-values.  The choice of an appropriately-chosen display can be important to visually verify, beyond the indicated color-coding, 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 y-values to the plotted range of x-values can vary. For example, if the plotted range of y-values is from −10 to 10 and the plotted range of x-values 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 y-values to the plotted range of x-values (or vice-versa) when it is greater than 5 to 1.

Domain and Range

Domain. The domain of a function is the set of all permissible x-values. 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 y-values attained. The range of an odd degree polynomial is always all real y-values, −∞ < y < ∞, while the range of an even degree polynomial, excluding 0 degree polynomials, can be described using inequalities either in the form of −∞ < yglobal maximum, or global minimumy < ∞. Often, the range is expressed using interval notation, in which case the interval needs to be visually understood as a y-interval. 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 y-value, {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: yx and yx 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=x2 (↑↑: 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=−x2 (↓↓: 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 y-Axis. The graph of y=f (x) is symmetric with respect to the y-axis 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 y-value f (c) is a local maximum value (also called relative maximum value) of f if there is an open interval containing the x-value 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 y-value f (c) is a local minimum value (also called relative minimum value) of f if there is an open interval containing the x-value 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 real-valued 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 y-value 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 y-value 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)=x3, 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)=– x3, 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 real-valued 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

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 real-valued 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 real-valued 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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