Examples of Rational Expressions. Rational
expressions are ratios of polynomial expressions, like: Rational Functions. Rational
functions are ratios of polynomial functions. Strictly speaking,
any polynomial, 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.
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 y-Intercepts. Any
point where a graph intersects the y-axis is called a 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 Vertical
Asymptotes. Rational functions are distinguished, in
part, by the possibility of having vertical asymptotes. A vertical
asymptote is a vertical line, 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 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 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. 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 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 Determining
Local Maxima and Local Minima. Any value
of x in the domain of f ' for which Increasing. Geometrically
speaking, a function Decreasing. Geometrically
speaking, a function 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 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 Determining Intervals
of Concavity. For a rational 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
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