Rational Functions: Concepts & Procedures

Examples of Rational Expressions. Rational expressions are ratios of polynomial expressions, like: (6 x< – 2)/(5 x²– 17), (–41 x³ + 7)/ (2^1/2 x⁷–4), and the ratio 3 x(x² – π)/x.

Rational Functions. Rational functions are ratios of polynomial functions. Strictly speaking, any polynomial, y = f (x), is also a rational function since it can be expressed as the ratio of two polynomials, f (x)/1.

Complete Graph. A complete graph is one that includes all the important mathematical features. For rational functions, this means the viewing window should be just large enough to display all x- and y-intercepts, all maxima and minima, all points of inflections, all holes, all vertical asymptotes, and all polynomial asymptotic end behaviors (horizontal asymptotes, slant asymptotes, etc.). When you enter a rational 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 rational function.

Local Maxima and Minima Predicted by Color Coding Near the Origin

Local Maxima and Minima Exposed by Smart Zooming from Xmin=–0.2 to Xmax=0.5

Domain and Range

Domain. The domain of a rational function is the set of all permissible x-values. Values of x that make the denominator zero are not permitted, because division by zero is not defined.

x-Intercepts. Any point where a graph intersects the x-axis is called an x-intercept of the graph. The maximum number of x-intercepts is the degree of the polynomial numerator while the minimum number is none.

y-Intercepts. Any point where a graph intersects the y-axis is called a y-intercept of the graph. If x=0 is in the domain of a rational function, there is exactly one y-intercept; otherwise, there is none.

Range. The range of a rational function can be determined approximately from its graph. It can often be determined analytically for elementary cases. However, for general rational functions, the analytic determination can go beyond what is expected of students in a high school or college math course. Here, we compute and display the range in The Analyzer using analytic means, as long as the computation time is not excessive.

Vertical Asymptotes
(What are Asymptotes?)

Vertical Asymptotes. Rational functions are distinguished, in part, by the possibility of having vertical asymptotes. A vertical asymptote is a vertical line, x = c, that the graph of the rational function approaches but never touches. For rational functions, vertical asymptotes always occur at the zeros of the denominator that are not eliminated by improper cancellation. Thus, x = c is a vertical asymptote if x – c is a factor of the denominator that cannot be eliminated by improper cancellation. As x approaches such a denominator zero from either side, the y-values of the function tend to either +∞ or −∞. This is customarily indicated on the graph by displaying the vertical asymptote as a dashed vertical line. The curve never makes contact with the dashed vertical line, since the defining rational function does not have any y-value at the denominator zero. Vertical asymptotes are sometimes referred to as essential singularities, since no redefinition of the function can ever make it continuous at the vertical asymptote (for contrast, see holes below). Examples of vertical asymptotes can be seen as the two dashed vertical lines in the left graph above.

Holes

Holes. When a rational function has a common factor in the numerator and denominator that can be eliminated in the denominator by improper cancellation, the importance of this lies in the fact that the rational function is not defined at the zero of the common factor since it appears in the denominator. Any common factor that is eliminated in the denominator due to improper cancellation does not give rise to a vertical asymptote. This is traditionally signified on the graph by having an empty circle appear at the place where the function would otherwise be defined. Such holes are sometimes referred to as removable singularities (or removable discontinuities), since, unlike vertical asymptotes, the rational function could be redefined at that point to make the redefined function continuous at that point. Here is an example of a rational function with a hole at x=1.

Other Asymptotes and End Behaviors

Asymptotic End behaviors. If you divide the numerator of a rational function by its denominator, you get a polynomial quotient (by itself, a polynomial) plus a polynomial remainder term (also a polynomial) whose degree is less than that of the denominator. The latter, when divided by the denominator, contributes very little to the y-values of the rational function for large |x|. For this reason, we say that the rational function is asymptotic to the polynomial quotient. The end behaviors of the rational function follow the end behaviors of the polynomial quotient. For example, the rational function (x³− 1)/(x²−2) can be expressed (using polynomial division) as x+(2x− 1)/(x²−2). Here, the polynomial quotient is x so the rational function is asymptotic to y=x and, as a consequence, the end behaviors are down on the left and up on the right.

Linear Asymptotic End Behaviors: Horizontal and Oblique (Slant) Asymptotes. Special names are given to low degree polynomial quotients. If the degree of the polynomial quotient is 0, the polynomial quotient is some constant, k, and the rational function is said to have the horizontal asymptote y=k. The graph above has a horizontal asymptote, y=0, shown as a dashed line (the x-axis). Unlike a vertical asymptote, the rational function may, but need not, intersect the horizontal asymptote. If the degree of the polynomial quotient is 1, the polynomial quotient is a linear function and the rational function is said to have an oblique (or slant) asymptote equal to that linear function. Again, the rational function may, but need not, intersect the linear asymptote.

Higher Degree Polynomial Asymptotic End Behaviors. There can also be quadratic asymptotic end behaviors, cubic asymptotic end behaviors, and so forth. For these reasons, we refer to the polynomial quotient (regardless of low or high degree) as a polynomial asymptote, and every rational function has a polynomial asymptote. In general, the rational function may, but need not, intersect its polynomial asymptote. On WebGraphing.com, the polynomial asymptotes are shown as dashed polynomial curves. Traditionally, only horizontal or slant asymptotes (degrees 0 and 1) are included on graphs of rational functions, but here we include all polynomial asymptotes regardless of degree for their value in reading the graph of a rational functions. Note: While a rational function may have many or no vertical asymptotes, it has exactly one polynomial asymptote that is useful for clarifying its end behaviors.

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 continuous 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 continuous 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 in the domain of f ' for which f '(x) = 0 is called a zero of f '. For rational functions, the local maxima and local minima can only occur at the real-valued zeros of f '. These zeros, together with the x-values where f ' is not defined (the denominator zeros of f ), 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, which are described in Polynomial Functions: Concepts and Procedures.

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 rational function f (x) = x⁵/( x²+1), 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 rational function f (x) = –x⁵/(x²+1), 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 rational function y = f (x), first find all the real-valued zeros of f '. These zeros, together with the x-values where f ' is not defined (the denominator zeros of f ), 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 the 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 ', undefined numbers, 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. Any point not in the domain cannot be a point of inflection. Can you think of an example where (1) is true but (2) is not true?

Determining Points of Inflection. Any value of x in the domain of f '' for which f ''(x) = 0 is called a zero of f ''. For rational functions, a point of inflection can only occur at a real-valued zero of f ''. These zeros, together with the x-values where f '' is not defined (the denominator zeros of f ), separate the real number line into intervals. To determine the 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, which are described in Polynomial Functions: Concepts and Procedures.

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