Complete function graphing system and method

[0001] The present application is a continuation-in-part of the U.S. Utility application Ser. No. 11/273,495, filed on Nov. 14, 2005, entitled FUNCTION GRAPHING SYSTEM AND METHOD. The present application claims priority to U.S. Provisional Application No. 60/630,697 filed Nov. 23, 2004 entitled FUNCTION GRAPHING SYSTEM AND METHOD which is incorporated herein in its entirety by reference.

[0002] The present invention relates generally to function graphing and, more specifically, to automatic determination of optimum graph characteristics.

[0003] Mathematical functions can be classified as either algebraic or transcendental (nonalgebraic). Algebraic functions are expressions that involve only the algebraic operations of addition, subtraction, multiplication, division, raising to an integer power, and extracting an odd number root or even number root. Elementary transcendental functions include exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic trigonometric, inverse hyperbolic trigonometric, tower functions, as well as combinations of these functions together with algebraic (including polynomial and rational) functions into piecewise-defined functions. In terms of graphing, a key difference that distinguishes both algebraic functions and transcendental functions that do not include any of the six trigonometric functions from other transcendental functions is that the former functions have a finite number of points of mathematical interest such as: intercepts, extrema, points of inflection, holes, vertical asymptotes, horizontal asymptotes, slant asymptotes, points of discontinuity, isolated points, vertical tangents, vertical cusps, and corners. This means that graphs of algebraic functions and transcendental functions that do not include any of the six trigonometric functions are capable of being displayed in a complete graph—one that includes all their points of mathematical interest. Other transcendental functions, on the other hand, including the six trigonometric functions, can have an infinite number of points of mathematical interest.

[0004] Software packages and many graphing calculators use algorithms that could display complete graphs of functions if the selection of function-specific graphing settings such as x-dimension values and y-dimension values were appropriate. Further, if all the important mathematical features, together with standard auxiliary graphics for illuminating singularities (holes, discontinuities, isolated points, and the various kinds of asymptotes), could be provided for in the function-specific settings, significant features could be more clearly apparent.

[0005] What is needed is a system and method to provide an electronic graph of a function in which important mathematical features of the function, including singularities, are clearly apparent, for example, but not limited to, within an optimum viewing window, for example, but not limited to, on a graphing calculator, a personal computer display, a handheld computer, or other type of electronic display. Further, a system and method are needed to provide an efficient way for a user to view mathematical features of a user-selected function. Still further, the use of traditional auxiliary graphics for illuminating singularities (dashed lines to designate asymptotes, small empty circles to designate excluded curve endpoints, and small filled circles to designate included curve end points or isolated points) can help users to better identify unusual features of a given function from its graph, and, moreover, mathematical analyses of a given function reported with numerical results in both exact (symbolic) forms, which users typically get when obtaining solutions by hand, as well as approximate forms can help users to review calculation results.

[0006] The problems set forth above as well as further and other problems are solved by the present invention. These solutions and other advantages are achieved by the various embodiments of the invention described herein below.

[0007] The present invention includes a system and method for optimally graphing functions. The graphing device for enabling the system and method of the present invention may be any type of electronic display, for example, a graphing calculator. The method of the present invention can include, but is not limited to, the steps of accepting a function, error-checking each accepted user entry, determining the type of function entered, determining the domain of the function entered, determining mathematical features of interest with respect to the accepted function, and determining an optimum viewing window that can include all the mathematical features of interest when there are, at most, finitely many such features. In addition, the method can include the step of creating auxiliary graphics to illuminate singularities on the graph. The method can further include the steps of highlighting particular curve segments, for example by color-coding them, to assist visual analysis, and plotting the highlighted curve segments in the optimum viewing window. Optionally, the method can include the steps of resizing the optimum viewing window upon user request, preparing tabular displays for numerical values of important mathematical features in both approximate and symbolic (exact) values, when appropriate, and displaying appropriate scales on the graph.

[0008] The system of the present invention can include, but is not limited to, a user entry subsystem, a features subsystem, and a window subsystem, which together can implement the method of the present invention. The system of the present invention can optionally include subsystems (1) to customize the default values in the system and (2) to check for entry errors during function processing.

[0009] The highlighted graphs of the present invention can help users to better visualize the properties of increasing/decreasing and concave up/concave down for a given function, each property highlighted on a separate graph. In addition, the two graphs can be presented, optionally, to transform from one to the other, or to toggle between the two graphs, to help users visually to compare and distinguish between the properties being highlighted. Also, peaks and valleys (extrema) can be inferred at the junctures between increasing/decreasing segments that are highlighted in a way that distinguishes those segments from one another. Similarly, changes in concavity can be inferred at the junctures between concave up/concave down segments that are highlighted in a way that distinguishes them from one another. Resizing in the vicinity of the juncture when a graph is flat, entering only x-dimension values and permitting automatic determination of optimum y-dimension values, can enable the user to visually verify the existence of a peak or valley as well as a change in concavity.

[0010] The system of the present invention can use traditional auxiliary graphics for illuminating singularities (dashed lines to designate asymptotes, small empty circles to designate excluded curve endpoints, and small filled circles to designate included curve end points or isolated points) that can help users to better identify unusual features of a given function from its graph. In the system of the present invention, mathematical analyses of a given function are reported with numerical results in both exact (symbolic) forms, which users typically get when obtaining solutions by hand, as well as approximate forms.

[0011] For a better understanding of the present invention, reference is made to the accompanying drawings and detailed description. The scope of the present invention is pointed out in the appended claims.

[0032] The present invention is a system and method for optimally graphing: (1) algebraic functions and transcendental functions that have finitely-many points of mathematical interest, obtaining a complete graph, and (2) transcendental functions that have infinitely-many points of mathematical interest, including the trigonometric functions. All examples provided herein are illustrative of aspects of the invention and not for limiting the uses of the invention.

[0033] Referring now to FIG. 1, system 10 can include, but is not limited to, user entry subsystem 23, features subsystem 27, and window subsystem 31. Optionally, system 10 can include resize subsystem 35, customizer 25, function/type subsystem 29, and error checking subsystem 33. System 10 can execute in a variety of environments, from a calculator to at least one computer node 26 within a communications network 21. System 10 can interoperate with display 19, which can interface with CPU 14 that can execute system 10. CPU 14 can also interconnect with communications network 21 through electronic interface 11. Window subsystem 31 can display graph 20 within window 22 on display 19, and/or save graph 20 on at least one computer readable medium 15, which can also interface with CPU 14 through electronic interface 11 and/or through communications network 21.

[0034] Continuing to refer to FIG. 1, operationally, user 13 and/or administrator 17 can provide information through user entry subsystem 23. The information can be used by system 10 to build, highlight, and display graph 20. User 13 can provide function 16 through user entry subsystem 23 to error checking subsystem 33, and in turn, to function/type subsystem 29. Optionally, administrator 17 can provide additions/updates to system defaults such as, for example, type 18 and highlighting 38. After function 16 has been entered and type 18 determined by function/type subsystem 29, features subsystem 27 can determine at least one mathematical feature 24 and provide it to window subsystem 31. Window subsystem 31 can determine, from the at least one mathematical feature 24, the dimensions 34 of graph 20 that would include as many as all the at least one mathematical features 24. Window subsystem 31 can also determine at least one curve segment 36, if possible, and select appropriate highlighting 38 for the at least one curve segment 36, if one is found. When window subsystem 31 is unable to determine at least one curve segment 36, for example because calculations can take too long, window subsystem 31 can provide, for example, a single color curve that can include other mathematical features such as, for example, asymptotes and holes (FIG. 13). Window subsystem 31 can provide graph 20, dimensions 34, at least one curve segment 36, if possible, and highlighting 38 to display 19, which can display window 22 containing graph 20. Customizer 25 can allow administrator 17 to add or update, for example, but not limited to, types 18 and/or highlighting 38. Function/type subsystem 29 can determine type 18, for example, but not limited to, polynomial, rational (but not polynomial; hereinafter referred to as rational), algebraic (but neither polynomial nor rational, hereinafter referred to as algebraic or general algebraic), and transcendental. Function/type subsystem 29 can also provide an association between type 18 and at least one mathematical feature 24. Error checking subsystem 33 can check the syntax of function 16, and can inform user 13 of errors. Error checking subsystem 33 can also provide at least one report 42 of at least one error to user entry subsystem 23. Continuing to further refer primarily to FIG. 1, system 10 can process entered functions 16 as follows. For polynomial functions, window subsystem 31 can determine an optimum viewing window 22 with dimensions 34. All points corresponding to zeros of ƒ, ƒ′ and ƒ″ are included in the optimum viewing window 22. This is accomplished by finding all pairs of x- and y-coordinates for up to all the important at least one mathematical features 24: intercepts, extrema, and points of inflection. These coordinates can be fit into a viewing window 22 above and below the maximum and minimum x- and y-coordinates by, for example, 10% of the smallest window 22 that includes all the important at least one mathematical features 24. If there is only one such pair of coordinates (there must be at least one for the y-intercept), a convenient value, for example, 1 unit above and 1 unit below can be selected. For polynomials, the domain 32 is always (−∞, ∞). Range 28 of polynomial function 16 can be determined based on the degree of the polynomial; for example, odd degree polynomials have a range 28 of (−∞, ∞). To determine the form of the range 28, window subsystem 31 can determine the sign of the leading coefficient of the polynomial, a_n, and can perform the following tests: (1) if the leading coefficient is positive, range 28 can be in the form [ƒ(x_m), ∞), where ƒ(x_m) is the global minimum value amongst all values in the set {ƒ(x_m)}, where ƒ′(x_m)=0; (2) if the leading coefficient is negative, range 28 can be in the form (−∞, ƒ(x_m)], where ƒ(x_m) is the global maximum value amongst all values in the set {ƒ(x_m)}, where ƒ′(x_m)=0. The plotting interval of x-coordinates can be adjusted so that, for example, the curve can extend across the display 19. The resulting viewing window 22 is optimum in the sense that it can include all the important at least one mathematical features 24 of a polynomial function 16 (intercepts, extrema, and points of inflection), can extend the curve horizontally across the display 19, and does not flatten out peaks and valleys unnecessarily by having too large a plotting y-dimension 28A.

[0035] Continuing to still further refer to FIG. 1, to determine an optimum viewing window 22 for a rational function, window subsystem 31 can find all pairs of x- and y-coordinates for all the important at least one mathematical features 24: intercepts, extrema, points of inflection, and holes (FIG. 13). Also, when vertical asymptotes are present, additional coordinates are included in the viewing window where the slope of the function in absolute value is large, say for example, 81. Further, when a horizontal asymptote is present, additional coordinates can be included in the viewing window where the slope in absolute value is relatively small, for example, 1/50. When a slant or quadratic or cubic or other higher degree polynomial asymptote is present, additional coordinates are included in the viewing window to display the important features of that polynomial asymptote as, for example, a dashed polynomial curve in the viewing window. Additional analyses can determine if improper cancellation creates anomalies. For example, the expression

[0036] $\frac{x-1}{x-1}$
could be improperly evaluated as a value of 1, whereas the correct evaluation is the constant function 1 with a hole at the coordinate (1, 1). The domain 32 of a rational function 16 is all real numbers except for zeros of its denominator. The range 28 of a rational function 16 can be determined by window subsystem 31 as follows: determine all denominator zeros, split the number line into subintervals where endpoints are either +∞, −∞, or a denominator zero. On each such subinterval, the curve is unbroken and the range is of the form (−∞, a), (−∞, a], [a, b], (a, b], [a, b), (a, b), (a, ∞), [a, ∞). By determining asymptotic end-of-interval behaviors and local extrema, the range of each unbroken curve can be determined. The range of the rational function is then the union of the ranges of the unbroken curves.

[0037] Continuing to still further refer to FIG. 1, to determine an optimum viewing window 22 for an algebraic function or an transcendental function that does not include any of the six trigonometric functions, in addition to the coordinates found in the case of rational functions, window subsystem 31 can also include coordinates for any finite domain endpoints and isolated points (FIG. 14) when isolated points are present, jump discontinuities (FIG. 13) when jump discontinuities are present, vertical cusps (FIG. 15) when they are present, vertical tangents (FIG. 12) when they are present, and corners (FIG. 13) when they are present. Note that for general algebraic or transcendental functions, it is possible that there are no important mathematical features at all. This could happen, for example, if the function ƒ(x)=√{square root over (x−1)}+√{square root over (2−x)} were entered since it has no domain 32; that is, there is no value of x for which the function is real, so there should be no graph 20 delivered. (Strictly speaking, the empty set is a function.) Other anomalies include having only a finite number of isolated points (no curve) to plot, as is the case, for example, when ƒ(x)=2+√{square root over ((x²−1)(1−x²))}{square root over ((x²−1)(1−x²))} which is real-valued only when x=±1. In this case, the only points to plot are isolated points with coordinates {(−1,2), (1,2)}. Additional analyses can determine if improper cancellation creates anomalies. For example, the expression

[0038] $\frac{\sqrt{x^2-1}}{\sqrt{x^2-1}}$
could be improperly evaluated as a value of 1, whereas the correct evaluation is the constant function 1 for |x|>1, with holes at the coordinates (−1, 1) and (1, 1). The domain 32 of an algebraic function 16 is all real numbers except for zeros of its denominator and negative values of radicands of even roots. The domain 32 of a transcendental function 16 is function-specific and depends on the precise definition of each such function and its argument. All elementary transcendentals are included in window system 31.

[0039] Referring now primarily to FIG. 2, user entry subsystem 23 can include, but is not limited to, function receiver 63 and custom receiver 62. Function receiver 63 can process any type 18 of function 16. Four types 18 of functions 16—polynomial, rational (FIG. 11), algebraic. (FIGS. 12-15), and transcendental functions (FIG. 16)—are described herein, but the invention is not limited to these types 18. Function receiver 63 can process polynomial functions, which are expressions that involve only the algebraic operations of addition, subtraction, multiplication, and raising to a non-negative integer power. Function receiver 63 can also process rational functions. Rational functions can be distinguished from polynomial functions by the appearance of asymptotes (vertical, horizontal, and slant) and holes (FIG. 11). Function receiver 63 can also process algebraic functions, which are expressions that involve only the algebraic operations of addition, subtraction, multiplication, division, raising to an integer power, and extracting an odd number root or an even number root. Taking roots—the operation that distinguishes general algebraic functions from either polynomials or rational functions—can affect function processing in at least two ways: (1) taking even roots may reduce the size of a function's domain 32 considerably, possibly limiting the display region, while (2) taking odd roots may create distinctive visual characteristics that are not possible with either polynomial or rational functions. General algebraic functions can be distinguished from either polynomials or rational functions by the appearance of vertical tangents (FIG. 12), vertical cusps (FIG. 15), corners, two-hole jump discontinuities (FIG. 13), and isolated points (FIG. 14). Function receiver 63 can also process transcendental functions, which are functions that cannot be constructed solely from algebraic operations. Transcendental functions can have all the mathematical features that algebraic functions have, as well as one-hole jump discontinuities, and the properties of being periodic or having amplitude. Custom receiver 62 can receive and process changes in various parameters upon which system 10 depends. For example, a new type 18 can be defined through custom receiver 62 and customizer 25. User entry subsystem 23 can notify, for example, function receiver 63 that a new type 18 has been defined. Function receiver 63 can determine if a function is a single-formula function or a piecewise function and can route the function to either single-formula processor 63A or piecewise processor 63B. Piecewise processor 63B determines behavior and auxiliary graphics needed at the juncture points of the piecewise function where pieces meet, and processes each piece in a manner similar to transcendental function processor, regardless of type. Piecewise processor 63B does not display the final curve until after the last piece is processed, combining at the end the piecewise graphs from all pieces together with the additional auxiliary graphics determined at the juncture point as well as all the auxiliary graphics associated with the pieces, and delivering two optimal highlighted graphs. If the various pieces exclude the trigonometric functions, the resulting graphs include all the points of mathematical interest for the entire piecewise function: they are complete graphs.

[0040] Referring now primarily to FIG. 3, features subsystem 27 can include, but is not limited to, intercept finder 65, extrema finder 69, points of inflection finder 73, holes finder 77, corner finder 81, horizontal/vertical asymptote finder 71, vertical tangent finder 75, vertical cusp finder 79, one-hole and two-hole jump discontinuity finder 83, symmetry finder 66, period finder 68, amplitude finder 72, range finder 78, domain finder 74, and isolated points finder 76. Values like the square root of 2 are commonly referred to as symbolic or exact values, while the corresponding approximate value, 1.414, is referred to as a numerical value. Certain symbolic form values can be useful to see, such as the symbolic form values that a user would typically get when solving an equation by hand. Features subsystem 27 can automatically display these symbolic forms in all tables created by features subsystem 27 and in interval notation answers. More specifically, each of the modules of features subsystem 27 listed above is described below.

[0041] To find all x-intercepts, intercept finder 65 can solve the equation ƒ(x)=0 both symbolically (exactly), when possible (for polynomials, mathematical theory only guarantees symbolic solutions for polynomials of degrees 4 or less), and numerically. Any values that are also denominator zeros (they cannot be intercepts) are excluded. To find all y-intercepts, when x=0 is not a denominator zero, intercept finder 65 computes x=0 and y=ƒ(0).

[0042] To find all extrema, extrema finder 69 can solve the equation ƒ′(x)=0 both symbolically (exactly), when possible, and numerically. Numbers in the set {x_m} of numbers that satisfy this equation (the zeros of ƒ′) are possible x-coordinates of local maxima and local minima, so the points in the set {(x_m, ƒ(x_m))} become the extrema. A table of values can be constructed to help determine local maxima and minima. The table can consist of the solutions, x_m, from above, together with conveniently selected “test values” (integer values, whenever possible) between and beyond these numbers, which indicate whether the function is increasing (the first derivative is positive) or decreasing (the first derivative is negative) between values of x_m. The table can have up to, for example, 6 columns, including values for x (symbolic (exact), if available and not too lengthy, as well as numerical), ƒ(x) (symbolic (exact), if available and not too lengthy, as well as numerical), ƒ′(x) (numerical), and the sign of ƒ′(x) or an indication, when ƒ′(x)=0 and has no sign or is undefined, of whether it is a local maximum or minimum or not. Using the information given in the table of local maxima and minima described above, the at least one curve segment 36 where the function is increasing and decreasing can be determined, displayed, and highlighted.

[0043] To find all holes (FIG. 13) and horizontal/vertical asymptotes, holes finder 77 and horizontal/vertical asymptote finder 71 determine all denominator zeros by setting the denominator, den(x), equal to zero and solving the equation den(x)=0. If a denominator zero, say x=a, can be eliminated due to improper cancellation, it is the x-coordinate of a hole (FIG. 13); otherwise, x=a is the equation of a vertical asymptote. Horizontal/vertical asymptote finder 71 further investigates vertical asymptotes from each side to determine if the limiting behavior is +∞ or −∞, and this information can be reported to graph creator 93, which can prepare for example, a table containing the information. To find any horizontal asymptotes in a rational function, horizontal/vertical asymptote finder 71 can compute

[0044] $\underset{x\to \infty }{\lim }f\left(x\right).$
If this value is numeric, referred to as b, then y=b is the equation of a horizontal asymptote; otherwise, there is no horizontal asymptote. To find any horizontal asymptotes of a non-rational function, horizontal/vertical asymptote finder 71 can compute two limits:

[0045] $\underset{x\to \infty }{\lim }f\left(x\right)=b_1$
and

[0046] $\underset{x\to -\infty }{\lim }f\left(x\right)=b_2.$
Whenever either (or both) of b₁ and b₂ is finite, then either (or both) of b₁ and b₂ corresponds to a horizontal asymptote (in case both are finite and b₁≠b₂, there are two horizontal asymptotes: y=b₁ and y=b₂); otherwise, there is no horizontal asymptote.

[0047] To find all points of inflection (these denote changes in concavity), points of inflection finder 73 can solve the equation ƒ″(x)=0 both symbolically (exactly), when possible, and numerically. Numbers in the set {x_poi} of numbers that satisfy this equation (the zeros of ƒ″) are possible x-coordinates of points of inflection, so all points in the set {(x_poi, ƒ(x_poi)} are possible points of inflection. A table of values can be constructed to help determine concavity and points of inflection (FIG. 20). The table can consist of the solutions, x_poi, from above, together with conveniently selected “test values” (integer values, whenever possible) between and beyond these numbers, which indicate whether the function is concave up (the second derivative is positive) or concave down (the second derivative is negative) between values of x_poi. The table can have up to 6 columns, including, for example, values for x (exact (symbolic), if available and not too lengthy, and numerical), ƒ(x) (exact (symbolic), if available and not too lengthy, and numerical), ƒ″(x) (numerical), and the sign of ƒ″(x) or an indication, when ƒ″(x)=0 and has no sign or is undefined, of whether it is a point of inflection or not. Using the information given in the table of points of inflection described above, at least one curve segment 36 where the function is concave up and concave down can be determined, displayed, and highlighted.

[0048] Jump discontinuity finder 83 analyzes holes (FIG. 13) with regard to the left and right hand limits; if the limits are unequal but real-valued, that location is designated as a jump discontinuity. If the function is undefined at the jump, it is designated a two-hole jump (FIG. 13). If the function is defined at the jump (and is left- or right-continuous at the jump), it is designated as a one-hole jump (FIG. 16).

[0049] To find vertical tangents (FIG. 12), vertical cusps (FIG. 15), and corners (FIG. 13), Vertical tangent finder 75, vertical cusp finder 79, and corner finder 81 can rewrite any function involving the absolute value function (such a function is an algebraic function) using the alternative form, abs(x)=√{square root over (x²)} and can compute the first derivative. Vertical tangent finder 75, vertical cusp finder 79, and corner finder 81 can set the denominator of the first derivative equal to zero and solve for x to determine where there is no slope (where the first derivative is not finite). If the derivative is not defined at x=a, and if both

[0050] $\underset{x\to a^{+}}{\lim }{f}^{\prime }\left(x\right)$
and

[0051] $\underset{x\to a^{-}}{\lim }{f}^{\prime }\left(x\right)$
are either +∞ or are −∞, the graph has a vertical tangent (FIG. 12) at x=a. If both values are infinite but with different signs (one is +∞ and the other is −∞), x=a is a vertical cusp (FIG. 15). If x=a is a domain end point and the limit from the defined side is either +∞ or −∞, then the graph has a one-sided vertical tangent (FIG. 12) at x=a in the sense that the slopes of the tangent lines to the curve from one side approach +∞ or −∞.

[0052] To find isolated points (FIG. 14), isolated point finder 76 can analyze function 16 before any intervening simplifications are done that might cause information to be lost. During the analysis, isolated point finder 76 can find up to all values for which an even root radicand might be zero and isolated. Isolated point finder 76 can plot isolated points as, for example, solid small circles, designating in the standard way that a function 16 is defined at a point of singularity.

[0053] To find at least one mathematical feature 24 that is defined by, for example, user 13 interfacing with customizer 25, other feature finder 84 performs a process that can be defined by user 13, or through any other means, through customizer 25.

[0054] To find symmetry properties of function 16, symmetry finder 66 can determine, for example, if ƒ(−x)=ƒ(x), in which case the function is even, or if ƒ(−x)=−ƒ(x), in which case the function is odd; several techniques are used, depending on the type of function, including the technique of choosing a large number of randomly selected points in the domain, {x_i}, and comparing the values {ƒ(−x_i), ƒ(x_i)} and {ƒ(−x_i), −ƒ(x_i)} for equality to within a small numerical tolerance to determine with a high probability that the function does or does not possesses any symmetry property.

[0055] To find the period of function 16, period finder 68 computes, when trigonometric functions are present in the entered function, an appropriate “potential period” depending on the various arguments in the trigonometric functions and, further, to determine whether or not it is actually periodic, period finder 68 can, for example, choose, in the domain, a large number of randomly selected points, {x_i, x_i+p}, where p is the potential period as determined from the one or more arguments, and compare the values {ƒ(x_i), ƒ(x_i+p)} for equality to within a small numerical tolerance to determine with a high probability that the function is actually periodic with the computed “potential period” p. For example, x² sin(2x) has “potential period” of π due to the period of sin(2x) but it is not periodic due to the factor x².

[0056] To find the amplitude of function 16, amplitude finder 72 can, for example, compute, for sine and cosine functions when they are wrapped around other functions or arguments, the amplitude, as determined by taking one-half the maximum minus minimum value of the function.

[0057] To find the range of function 16, range finder 78 can, for example, compute, for a polynomial function the end behaviors and uses the global maximum or minimum, if there is one, to determine the domain, and, for a rational function, determines the end behaviors of the corresponding polynomial quotient and computes the range for each unbroken piece (pieces are separated by holes or vertical asymptotes) using any maxima, minima, and horizontal asymptote to make the determination, and then uniting all such ranges to obtain the range of rational function 16.

[0058] To find the domain of function 16, domain finder 74 can compute, for example, the domain by type: for polynomials, it is all real numbers; for rational functions, it is all real numbers excluding holes and vertical asymptotes; for algebraic functions, in addition to exclusions noted for rational functions, any jumps or negative radicands for even roots are excluded; and for transcendental functions, the precise domain for each function is defined and used in conjunction with the various arguments of the function to determine a final domain for the function.

[0059] Referring now primarily to FIG. 4, window subsystem 31 can include, but is not limited to, graph creator 93, dimension finder 85, curve segment finder 87, highlighter 91, behavior detector 89, and excluded values handler 102. For a polynomial function, graph creator 93 can display highlighted graph 20, which can include all the important at least one mathematical features 24 (FIG. 1) contained within the optimal plotting interval, including y-dimension 28A values (FIG. 1) that are optimized to show the locally extreme y-values restricted to the given plotting interval. Here, dimension finder 85 selects the y-dimension 28A values to expand the curve vertically to accentuate visually any peaks or valleys. When the location where the axes cross on a graph is not the origin, to make users aware of this fact, graph creator 93 can display a note under graph 20. Graph creator 93 can compute and display the ratio of the spread of y-dimension 28A values (upper minus lower) to the spread of x-dimension 32A values (upper minus lower) or the reverse, whichever is larger, which we briefly refer to here as the scale 39 (see FIGS. 11-15). Graph creator 93 can display the x- and y-intercepts, previously determined, in tabular form, with values given in symbolic (exact) form, if available and not too lengthy, as well as numerical form. Graph creator 93 can display vertical tangents (FIG. 12), vertical cusps (FIG. 15), and corners in tabular form and provide explanations for them.

[0060] Continuing to further refer primarily to FIG. 4, graph creator 93 can create graph 20 of a rational function 16 that can include, for example, up to all of its interesting mathematical features: x- and y-intercepts, local maxima and local minima, points of inflection, holes (FIG. 13), vertical asymptotes, well as the horizontal, or slant, or higher degree polynomial asymptote (quadratic, cubic, etc.) as the case may be. If there is a horizontal asymptote, for example, y=b, the numerical value b can be added to the set S_y of important y-values to be displayed; otherwise, features subsystem 27 can determine the locations of as many as all the important at least one mathematical features 24 of the requisite polynomial asymptote (as done when plotting a polynomial function) and all these locations are added to S_x (important x-values to be displayed) and S_y respectively, revising if necessary the maximum and minimum x-values from the “initial” plotting interval. Features subsystem 27 can determine the maximum and minimum of the y-values in S_y and can fit them into a viewing window 22 above and below the maximum and minimum x- and y-coordinates by some small percentage, such as for example 10%, of the smallest window 22 that includes all these points. If there is no x-value, i.e. when S_x is empty, the default minimum can be, for example, −1 and the default maximum can be, for example, +1. If there is only one x-value, features subsystem 27 can add a value, for example 1, to the only x-value to determine the maximum x-dimension, and can subtract a value, for example 1, from the x-value for the minimum x-dimension. Features subsystem 27 can determine y-value minima and maxima in the same way for the special cases where there is up to one y-value. The resulting viewing window 22 is optimum in the sense that it can include as many as all the important at least one mathematical features 24, such as, for example, vertical asymptotic and end behavior polynomial (horizontal, slant, etc.) asymptotic trends of a rational function and can highlight as many as all singularities.

[0061] Continuing further to refer primarily to FIG. 4, graph creator 93 can create graph 20 of a general algebraic or transcendental function 16 that can include, for example, as many as all of its interesting at least one mathematical features 24: x- and y-intercepts, local maxima and local minima, points of inflection, holes (FIG. 13), vertical and horizontal asymptotes, vertical tangents (FIG. 12), vertical cusps (FIG. 15), corners, jump discontinuities (FIGS. 13 and 16), and isolated points (FIG. 14). Highlighter 91 can show the singularities using highlighting 38 such as, for example, dashed lines in grey and either solid or empty small circles in an appropriate color (matching that of the curve at that location). In addition, if domain 32 has finite interval endpoints, highlighter 91 can show the end points of the curve, either as, for example, solid small circles when they are included or empty small circles when they are excluded. To graph functions with odd root powers, when the argument is negative, graph creator 93 can compute the real-valued root of the odd root power instead of the imaginary value that is its principal value. To do this, the sign of x can be multiplied by the root of the absolute value of x. This algorithm can provide a real-valued root for both positive and negative values of x. Further, graph creator 93 can display and provide the values for any horizontal and vertical asymptotes.

[0062] Continuing still further to refer primarily to FIG. 4, to further describe the determination of an optimum viewing window 22 for a rational function, ƒ, as previously outlined, dimension finder 85 can determine the set of x-coordinates, S_x, and the set of y-coordinates, S_y, for the following at least one mathematical feature 24: intercepts, extrema, points of inflection, and holes (FIG. 13). To further describe the determination of an optimum viewing window 22 for a general algebraic or transcendental function that does not include a trigonometric function, dimension finder 85 can determine the set of x-coordinates, S_x, and the set of y-coordinates, S_y for the following at least one mathematical feature 24: intercepts, extrema, points of inflection, isolated points, vertical and horizontal asymptotes, one-hole and two-hole jump discontinuities, discontinuities, vertical tangents, vertical cusps, and corners. In addition, if there are any horizontal asymptotes, dimension finder 85 can determine the respective coordinate locations where the slope becomes small regardless of sign (for example, solutions to

[0063] $f^{\prime }\left(x\right)=\pm \frac{1}{50}$
can be used to obtain the coordinates). Dimension finder 85 can further use the minimum and maximum x-coordinates in S_x for the “initial” plotting interval of x-values (subject to revision, as indicated below). If there are any vertical asymptotes, the respective coordinate locations where the slope is large regardless of sign can be used (for example, solutions of ƒ′(x)=±81 can be used to obtain the coordinates), as long as the corresponding x-values are within the “initial” plotting interval of x-values. Curve segment finder 87 can determine numbers in the set {x_m} of numbers, in addition to the denominator zeros, that satisfy the differential equation ƒ′(x)=0. Those numbers can split the plotting interval into at least one curve segment 36, if possible, where the graph is either increasing (when ƒ′(x)>0) or decreasing (when ζ′(x)<0), unless the function is constant. Alternatively, using the information given in the table of local maxima and minima discussed above, curve segment finder 87 can determine at least one curve segment 36, if possible, where a rational function is increasing and decreasing. Curve segment finder 87 can determine at least one curve segment 36, if possible, and graph creator 93 can plot function 16 on at least one curve segment 36. Highlighter 91 can highlight graph 20 in, for example, the color red on at least one curve segment 36 where function 16 is increasing, the color blue on at least one curve segment 36 where the function is decreasing, or the color black for a constant function (indicating that function 16 is neither increasing nor decreasing on any at least one curve segment 36). The one or more plots so obtained can be combined into a single graph 20 over the optimum viewing window 22 that displays function 16 with highlighting 38 (including, for example, the color black if function 16 is constant). Curve segment finder 87 can determine (and graph creator 93/highlighter 91 can plot) (1) holes (FIG. 13) (including, in the case of algebraic functions, two-hole jump discontinuities (FIG. 13)), that can be shown, for example, as empty small circles in highlighting 38, for example, the same color (red, blue, or black) as graph 20 at the location of the hole (FIG. 13); (2) any vertical or horizontal asymptotes that can be shown in highlighting 38 such as, for example, dashed lines in the color gray; (3) the slant or higher degree polynomial asymptote for rational functions that can be shown in highlighting 38 such as, for example, a dashed line in the color gray; (4) any isolated points (FIG. 14) as solid black small circles, and (5) end points belonging to a curve that can be highlighted as, for example, small circles in the color of the curve nearby.

[0064] Continuing yet still further to refer primarily to FIG. 4, curve segment finder 87 can determine numbers in the set {x_poi} of numbers, in addition to the denominator zeros, that satisfy the differential equation ƒ″(x)=0 (or using the information given in the table of points of inflection discussed above), and that can split the plotting interval into at least one curve segment 36, if possible, where the graph is either concave up (when ƒ″(x)>0) or concave down (when ƒ″(x)<0), unless function 16 is linear or constant. Graph creator 93 can plot function 16 on each such at least one curve segment 36, if possible, and highlighter 91 can highlight in, for example, the color purple, at least one curve segment 36 where function 16 is concave up, the color green on at least one curve segment 36 where function 16 is concave down, or the color black for a linear or constant function 16 (indicating that function 16 is neither concave up nor concave down on any at least one curve segment 36). The one or more plots so obtained can be combined into a single graph 20 over the optimum viewing window 22 that can display function 16 with highlighting 38 such as, for example, the colors purple and green (or, in the case of a linear or constant function 16, in, for example, the color black). Also, highlighter 91 can show (1) any holes (FIG. 13), for example, as empty small circles in the same highlighting 38, for example purple, green, or black, as graph 20 at the location of the hole; (2) any vertical (or horizontal, in the case of algebraic functions) asymptotes as, for example, dashed lines highlighted in the color gray; (3) the horizontal or slant or higher degree polynomial asymptote as, for example a dashed line highlighted in the color gray; and (4) any isolated points (FIG. 14) and end points belonging to a curve as solid black small circles. To avoid having solid vertical lines that are not part of the graph (they otherwise appear in plots that have vertical asymptotes and/or jump discontinuities), functions are plotted on at least one curve segment 36 where these values are endpoints, and the plots can then be combined at the end with dashed vertical lines for vertical asymptotes (FIGS. 11 and 20) and other auxiliary graphics for jump discontinuities (FIGS. 13 and 16). For general algebraic and transcendental functions, when the zeros of ƒ′ cannot be determined, highlighter 91 can replace an increasing/decreasing color-coded graph or multi-level grey-scale graph by a single color graph, for example, in black. Also, when the zeros of ƒ″ cannot be determined for general algebraic and transcendental functions, highlighter 91 can replace a concave up/concave down color-coded graph or multi-level grey-scale graph by a single color graph, for example, in black. Further, highlighter 91 can deliver a message to indicate the reason why the single color graph is presented.

[0065] Continuing further to refer primarily to FIG. 4, for polynomials, behavior detector 89 can determine end behaviors, according to theory, based on the leading term, a_nxⁿ. Behavior detector 89 can also determine the sign of a_n, and whether or not n is even or odd. In particular, (1) if a_n>0 and n is even, the behavior is up on the left and up on the right; (2) if a_n<0 and n is even, the behavior is down on the left and down on the right; (3) if a_n>0 and n is odd, the behavior is down on the left and up on the right; (4) if a_n<0 and n is odd, the behavior is up on the left and down on the right. The plotting region reported can consist of the dimensions 34 of the viewing window 22: minimum and maximum x-coordinates plotted, and minimum and maximum y-coordinates plotted. Behavior detector 89 can display this information to clarify and confirm what the user can read from the graph. For rational functions, behavior detector 89 can determine end behaviors based on computing the polynomial quotient, applying the above rules for polynomials, and displaying the results. For all types 18 of functions, behavior detector 89 can compute ƒ(−x) and perform the following symmetry tests: (1) determine if ƒ(x)=ƒ(−x), in which case the function is even and the graph is symmetric with respect to the y-axis, or (2) determine if ƒ(−x)=−ƒ(x), in which case the function is odd and the graph is symmetric with respect to the origin. Behavior detector 89 can report these results in tabular form. For trigonometric functions, behavior detector 89 can determine if the function is periodic, in which case it can compute and display the period, and when appropriate, behavior detector 89 can compute and display the amplitude.

[0066] With still further reference primarily to FIG. 4, excluded values handler 102 can display the excluded x- and y-intercepts, previously determined, for example, in tabular form, and can display denominator zeros (the x-coordinates) that give rise to holes (FIG. 13). Excluded values handler 102 can also investigate vertical asymptotes from each side to determine if the limiting behavior is +∞ or −∞, and can report this information in a table. For general algebraic and transcendental functions, excluded values handler 102 can determine the excluded end points of at least one curve segment 36 of increase and decrease from denominator zeros. Otherwise, the endpoints can be included, which is in accord with the definitions of increasing and decreasing. Denominator zeros can be taken into account to determine the excluded end points of concave up and concave down at least one curve segment 36. Otherwise, the endpoints can be included, which is in accord with the basic definitions of concavity segments (all points for which the chords connecting any two points all lie on one side of the curve). Excluded values handler 102 can determine if the value, x_poi, is a hole (FIG. 13) and needs, for this reason, to be excluded as a point of inflection, or if the value, x_m, is a hole (FIG. 13) and needs, for this reason, to be excluded as a local extrema.

[0067] Referring now primarily to FIG. 5, resize subsystem 35 can include, but is not limited to, x-dimension resizer 101 and x/y-dimension resizer 103. Resize subsystem 35 can access x-dimension 32A values of type 18, which are, for example, user-supplied. Optionally, user 13 can also supply y-dimension 28A values of resize subsystem 35. x-dimension resizer 101, which can help user 13 (FIG. 1) (through user entry subsystem 23) to resize the viewing window 22, can allow the entry of x-dimension 32A values (FIG. 1) alone (the lower and upper limits on x-coordinates, comprising the plotting interval). x-dimension resizer 101 can determine optimum y-dimensions 28A values (the lower and upper limits on y-coordinates) so that the viewing window 22 can include, for example, as many as all the important at least one mathematical features 24 of the function as described previously by type (intercepts, extrema, points of inflection, holes (FIG. 13), isolated points (FIG. 14), vertical asymptotes, as well as the horizontal or slant or higher degree polynomial asymptote, etc.) restricted to the infinite vertical strip determined by the user-specified plotting interval. Alternatively, x/y-dimension resizer 103 can allow the entry of both x-dimension 32A and y-dimension 28A values for full user control of the actual viewing window 22 dimensions 34. The result is a user-specified viewing window 22, containing highlighted graphs 20 to the extent that they lie inside this viewing window 22. Resizing in the vicinity of the juncture when the graph is flat, entering only x-dimension 32A values and permitting automatic determination of optimum y-dimensions 28A values can enable the user to visually verify the existence of the peak or valley or change in concavity. Resize subsystem 35 can be indirectly invoked by other actions taken by the user, for example, but not limited to, a mouse click-and-hold on at least one curve segment 36 can allow the user to “slide” along the curve in order to dynamically view further aspects of function 16.

[0068] Referring now primarily to FIG. 6, customizer 25 can include, but is not limited to, highlighter adder/updater 109, type adder/updater 105, features adder/updater 107, user entry adder/updater 112, and system adder/updater 114. Customizer 25 allows modifications to system 10. Type adapter/updater 105 can allow a user 13 or administrator 17 to add types 18 to system 10 and to update existing types 18. Highlighter adder/updater 109 can allow a user 13 or administrator 17 to update highlighting 38 preferences. Features adder/updater 107 can allow a user 13 or administrator 17 to add or update at least one mathematical feature 24 to system 10. User entry adder/updater 112 can allow user 13 or administrator 17 to add/update, for example, user entry prompts and other display options that, for example, window subsystem 31 present to user 13. System adder/updater 114 can allow user 13 or administrator 17, with appropriate authentications, to add/update internal parameters of system 10, for example, default values for resize subsystem 35.

[0069] Referring now primarily to FIG. 7, error checking subsystem 33 can include, but is not limited to, syntax checker 125 and other error checker 125A. Syntax checker 125 can check the user input for syntax errors, such as, for example, mismatched parentheses and symbols that are not permitted such as, for example, “!”. Syntax checker 125 can further cause an alert message to be delivered to the user indicating the nature of any syntax error. Other error checker 125A can check for non-syntax errors and report them to the user.

[0070] Referring now primarily to FIG. 8, function/type subsystem 29 can include, but is not limited to, function/type associater 111, type/feature associater 113, and type resolver 115. Function/type subsystem 29 can associate a type 18 with a function 16, where the type 18 determines which at least one mathematical feature 24 is associated with the function 16. Function/type associater 111 can allow a user 13 or administrator 17 to associate a certain function 16 with a type 18. Type resolver 113 can determine which type 18 is associated with function 16.

[0071] Referring now primarily to FIG. 9, method 50 of the present invention can include, but is not limited to, the steps of receiving function 16 (FIG. 1) (method step 131), such as a single formula function or a piecewise function and locating at least one mathematical feature 24 (FIG. 1) in the function 16 (method step 133). Method 50 can also include the steps of determining dimensions 34 (FIG. 1) of a window 22 (FIG. 1) that can include at least one mathematical feature 24 (method step 135) and creating window 22 according to dimensions 34 (FIG. 1). Method 50 can further include the steps of determining at least one curve segment 36 (FIG. 1), if possible, according to at least one mathematical feature 24 (method step 139) and setting highlighting 38 (FIG. 1) of at least one curve segment 36 according to at least one pre-selected behavior (method step 141). Method 50 can still further include the steps of creating a graph 20 (FIG. 1) of function 16 including at least one curve segment 36, if possible, and highlighting 38 (method step 143), and plotting graph 20 in window 22 (method step 145).

[0072] Referring now primarily to FIGS. 10A-E, and specifically referring to FIG. 10A, method 60 of the illustrative embodiment of the present invention can include, but is not limited to, the steps of selecting whether to enter a single formula function or a piecewise-defined function, using for example, FIGS. 18A-C for entry of a single formula function, optionally in coordination with a computer keyboard, or FIG. 19, for example, for entry of a piecewise-defined function (decision step 301). If a single formula function is to be entered (decision step 301), method 60 can further include the step of receiving the single formula function through user entry subsystem 23 (FIG. 2) (method step 303). If the entered function is syntactically correct (decision step 309), method 60 can further include the steps of determining the type 18 of function 16 (method step 315) and processing the particular type 18: for example, polynomial (method step 319), rational (method step 321), algebraic (method step 323), and transcendental (method step 325). Note that the invention is not limited to the types: polynomial, rational, algebraic, or transcendental functions. If the syntax of function 16 is incorrect (decision step 309), method 60 can further include the steps of delivering a syntax error message (method step 307) and returning to method step 303 to reenter function 16 for graphing and continue processing from there. If a piecewise-defined function is to be entered (decision step 301), method 60 can further include the step of receiving the piecewise-defined function through user entry subsystem 23 (FIG. 2) (method step 305). If the entered function is syntactically correct (decision step 313), method 60 can further include the step processing the piecewise-defined function (method step 317). If the syntax of piecewise-defined function 16 is incorrect (decision step 313), method 60 can further include the steps of delivering a syntax error message (method step 311) and returning to method step 305 to reenter piecewise-defined function 16 for graphing and continue processing from there.

[0073] Referring now to FIG. 10B, method 60 performs the processing indicated in method step 325 which can further include the steps of, for a transcendental function, if function 16 includes one of the six trigonometric functions (decision step 327), and if function 16 is periodic (decision step 329), determining the period and setting a flag to limit the x-dimension 32A values (method step 333). In particular, if the flag is set and the function is periodic, of length, for example, p, the x-dimensions 32A can be set, for example, to the interval [−p, p]. The dimensions of this graph are optimal in the sense that they serve to show graphically the periodic nature of the function together with all important at least one mathematical features 24 over the two periods (features that are being repeated indefinitely in both directions). If function 16 includes one of the six trigonometric functions (decision step 327), and if function 16 is not periodic (decision step 329), method 60 can include the step of setting the x-dimensions set to a fixed interval, for example, to the interval [−2π, 2π] (method step 331). If function 16 does not include one of the six trigonometric functions (decision step 327), method 60 continues processing at method step 173 (FIG. 10C).

[0074] Referring now primarily to FIG. 10C, method 60 can further include the steps of, for a polynomial function, determining coordinates of important at least one mathematical features 24 (FIG. 1) such as, but not limited to, intercepts, extrema, and points of inflection (method step 169); or for a rational function, determining coordinates of all important at least one mathematical feature 24 such as, but not limited to, intercepts, extrema, points of inflection, holes (FIG. 11), vertical asymptotes, and horizontal, slant, or other polynomial asymptote (method step 171); or for an algebraic or transcendental function, determining coordinates of all important at least one mathematical feature 24 such as, but not limited to, intercepts, extrema, points of inflection, holes (FIG. 13), isolated points (FIG. 14), horizontal and vertical asymptotes, vertical tangents (FIG. 12), vertical cusps (FIG. 15), corners, and jump discontinuities (FIGS. 13 and 16) (method step 173). Method 60 can further include the steps of determining optimum viewing window 22 (FIG. 1) dimensions 34 (FIG. 1) that include all important at least one mathematical features 24 (method step 175) and determining at least one curve segment 36 (FIG. 1), if possible, where function 16 (FIG. 1) is increasing and decreasing (method step 177). Method 60 can also include the step of plotting function 16 on each at least one curve segment 36, if possible, where function 16 is increasing in a color, for example, red, plotting function 16 on each at least one curve segment 36 where function 16 is decreasing in a color, for example, blue, and plotting function 16 where it is neither increasing nor decreasing in a color, for example, black (method step 181). Method 60 can still further include the step of combining the different color plots into a single graph 20 (FIG. 1), including auxiliary graphics for holes (FIG. 13), asymptotes, isolated points, and jump discontinuities (FIG. 16) (method step 185). Feature analysis continues as method 50 includes the steps of determining at least one curve segment 36, if possible, where function 16 is concave up and concave down (method step 179). Method 60 can also include the step of plotting function 16 on each at least one curve segment 36, if possible, where function 16 is concave up in a color, for example, purple, plotting function 16 on each at least one curve segment 36 where function 16 is concave down in a color, for example, green, and plotting function 16 where it is neither concave up nor concave down in a color, for example, black (method step 183). Method 60 can still further include the steps of combining the different color plots into a single graph 20 (FIG. 1), including auxiliary graphics, for example for holes (FIG. 13) and asymptotes (method step 187), displaying two optimum color-coded graphs if display flag is set (method step 191). If the user chooses not to resize window 22 (decision step 193), and if the user chooses not to enter a new single formula function 16 (decision step 189), method 60 includes the step of returning to method step 301 (FIG. 10A) to continue processing. If the user chooses to enter a new single formula function 16 (decision step 189) method 60 includes the step of resuming processing at method step 303 (FIG. 10A). If the user chooses to resize window 22 (decision step 193), method 60 continues processing at method step 195 (FIG. 10D).

[0075] Referring now primarily to FIG. 10D, if the user chooses to resize window 22 (FIG. 1) (decision step 193, FIG. 10C), and if there are no syntax errors in the user's entries (decision step 195), and if the user provides only x-dimension values 32A (FIG. 1) (decision step 199), method 60 can include the steps of determining coordinates of the important at least one mathematical features 24 (FIG. 1) within the limits of x-dimension 32A values entered (method step 201), and determining optimum viewing window 22 dimensions 34 (FIG. 1) in terms of y-dimension 28A values (FIG. 1) that include all important at least one mathematical features 24 (method step 205). If there are syntax errors in the user's entries (decision step 195), method 60 displays a syntax error message (method step 197) and tests further user input (decision step 195). If both x-dimension 32A and y-dimension 28A values are entered (decision step 199), method 60 can include the step of determining coordinates of the important at least one mathematical features 24 within limits of both x-dimension 32A and y-dimension 28A values (method step 203). In either case, if both x-dimension 32A and y-dimension 28A values are entered, or only x-dimension 32A values, method 60 can include the step of determining at least one curve segment 36, if possible, where function 16 (FIG. 1) is increasing and decreasing (method step 177, FIG. 10B et seq.).

[0076] Referring now to FIG. 10E, method 60 can further include the step of receiving a piecewise-defined function (method step 335). If there are syntax errors (decision step 339), method 60 can include the steps of providing an entry error message (method step 337) and resuming processing at method step 335. If there are no syntax errors (decision step 339), method 60 can include the step of determining behavior and auxiliary graphics needed at the juncture points where pieces meet (method step 341). If the first piece is being processed (decision step 343), method 60 can include the step of resetting the display flag (method step 345) and continuing processing at decision step 327 (FIG. 10B). If the first piece is not being processed (decision step 343), and a subsequent piece is being processing (decision step 347), method 60 can include the step of continuing processing at decision step 327 (FIG. 10B). Once all the pieces have been processed in this manner, method 60 can include the step of combining the piecewise graphs from all pieces together with the additional auxiliary graphics determined at the juncture point as well as all the auxiliary graphics associated with the pieces, delivering two optimal color-coded graphs. In other words, method 60 continues to process each piece through the previously described portal for transcendental functions, regardless of type, but not displaying the final curve until after the last piece is processed. If the various pieces exclude the trigonometric functions, the resulting graphs include all the points of mathematical interest for the entire piecewise function—they are complete graphs. If the user chooses to resize window 22 (decision step 351), method 60 continues processing at method step 195 (FIG. 10D). If the user does not choose to resize window 22 (decision step 351), and if a new piecewise function is to be entered (decision step 353), method 60 continues processing at method step 305 (FIG. 10A). Otherwise, method 60 continues processing at method step 301 (FIG. 10A).

[0077] Referring now primarily to FIGS. 11-15, highlighted, for example color-coded or gray-shaded, graphs 20 help users to better visualize the properties of increasing/decreasing and concave up/concave down for function 16. In addition, peaks and valleys (extrema) can be inferred at the juncture between highlighted, for example light gray and dark gray, or red and blue, segments of an Increasing/Decreasing graph, even if the graph is flat at the juncture. Similarly, changes in concavity can be inferred at the juncture between segments of a Concave Up/Concave Down graph, highlighted, for example, with the shades of light and dark gray, or the colors purple and green, even if the graph is flat at the juncture or there are no intervening turning points between consecutive points of inflection (FIG. 20). At least one curve segment 36 can be distinguished graphically through use of highlighting 38.

[0078] Referring now to FIGS. 16-19, exemplary graphs of transcendental (FIG. 16) and piecewise (FIG. 17) functions and exemplary calculator keyboards (FIGS. 18A-C) and piecewise entries (FIG. 19) are shown.

[0079] Referring now to FIG. 20, an exemplary algebraic function is shown having two horizontal asymptotes, one vertical asymptote, and a change in concavity when there is no intervening turning point (local extrema). The change in concavity is highlighted by the changes in grey scale and the locations are confirmed in an accompanying mathematical analysis that shows a table of values for the function and its second derivative.

[0080] Referring again primarily to FIG. 1, methods 50 and 60 (FIGS. 9 and 10A-E respectively) can be, in whole or in part, implemented electronically. Signals representing actions taken by elements of system 10 can travel over at least one live communications network 21. Control and data information can be electronically executed and stored on at least one computer-readable medium 15. The system can be implemented to execute on at least one computer node 26 in at least one live communications network 21. Common forms of at least one computer-readable medium 15 can include, for example, but not be limited to, a floppy disk, a flexible disk, a hard disk, magnetic tape, or any other magnetic medium, a CDROM or any other optical medium, punched cards, paper tape, or any other physical medium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, or any other memory chip or cartridge, or any other medium from which a computer can read. Further, the at least one computer readable medium 15 can contain graphs in any form including, but not limited to, JPEG and SVG.

[0081] Although the invention has been described with respect to various embodiments, it should be realized this invention is also capable of a wide variety of further and other embodiments. For example, the functions described herein, although not to be considered the only functions available for use within the system of the present invention, can be augmented by functions such as, for example, the greatest integer function, Airy functions, Bessel functions, and the Euler gamma function, without changing the scope of the invention.