Quotient Rule. The quotient a/b equals
0 if two conditions are met: (1) a=0 and (2) b≠0.
This zero quotient rule applies as well to quotients
of any two functions: the quotient
Functions Not Expressible as a Polynomial. Does a
rational expression like
Continuity and "Smoothness". Rational functions are continuous at every point in their domain. The domain excludes all denominator zeros (values of x that make the denominator equal to zero). Denominator zeros give rise to breaks in the curve, either holes or vertical asymptotes. As for smoothness, rational functions never have corners.
Vertical Asymptotes and Holes. Whenever
the denominator is zero at a particular value, say
Maximum Number of x-Intercepts. The numerator of a rational function is a polynomial and x-intercepts can only occur at zeros of the numerator (values of x that make the numerator equal to zero). So, as in the case of polynomial functions, the maximum number of x-intercepts equals the degree of the polynomial in the numerator.
Rational Function End Behaviors. Every rational function can be expressed as the sum of a polynomial quotient plus a fractional term in which the numerator has degree less than that of the denominator. For example,
Here, the polynomial quotient is (x+1)
and in the fractional term, the numerator, 2, has degree 0, which
is less than that of the denominator, x–1, which has
degree 1. The end behaviors of a rational function follow those
of its polynomial quotient, so in this example the end behaviors
follow those of
What are Asymptotes? Asymptotes are dashed lines or dashed polynomial curves that are displayed with the graph of rational and other nonpolynomial functions to help clarify certain types of limiting behaviors. Strictly speaking, asymptotes (whether vertical, horizontal, oblique, or some other polynomial curve) are not part of the function and it is for this reason that they are traditionally displayed as dashed lines, to distinguish them from the solid lines that represent the function. This manner of displaying asymptotes is standard for mathematicians and mathematics textbooks. Less sophisticated graphing calculators display the asymptotes either as a solid line or as no line at all, either of which can cause confusion with standard notation. On WebGraphing.com, asymptotes are always displayed as dashed.
Straight Line Asymptotes. A rational function may, but need not, have a vertical asymptote. Further, a rational function may be asymptotic to a nonvertical straight line in two ways. (1) If the degree of the numerator polynomial is less than or equal to the degree of the denominator polynomial, the rational function has a horizontal asymptote. (2) If the degree of the numerator polynomial exceeds by 1 the degree of the denominator polynomial, the rational function has a slant or oblique asymptote. These "rules" are quick ways to determine the existence of horizontal or slant asymptotes. To determine an asymptote explicitly, you need to compute the polynomial quotient.
Below is an example of a rational function with a horizontal asymptote. Note that the curve intersects the horizontal asymptote at (0,0). While it is impossible for a rational function to ever intersect any of its vertical asymptotes, this example illustrates how it is possible for a rational function to intersect its end behavior polynomial asymptote.
First, plot the y-intercept (set
For rational functions, the x-intercepts and vertical asymptotes are the only places where the y-value may, but need not, change sign. So, with one x-intercept at x=1 and one vertical asymptote at x=–2, the x-axis is split into three open intervals, (–∞,–2), (–2,1), and (1,∞), on each of which the function is either always positive (the graph is in the upper half-plane) or always negative (the graph is in the lower half-plane). Selecting convenient "test values" on each interval, say x=–3, x=0, and x=2, and substituting into the formula for y, we construct a table to determine the sign of y on each interval. The sign tells us whether the curve lies above or below the x-axis on each interval.
So far, there are no points plotted to the left of x=–2 but we can plot the test values from the table to remedy this and gain further insight. Also, using the information about the sign enables us to begin to sketch the graph nearby the plotted points:
To the left of x=–2, we can complete the graph by extending the curve so it is asymptotic to the vertical asymptote, x=–2, and the horizontal asymptote y=2. Note that on this half-plane, since the curve cannot cross the x-axis to the left of x=–2 (otherwise, the crossing would be another zero), there is only one way to approach the vertical asymptote: up (downward would mean crossing the x-axis). To the right of x=–2, we can complete the graph by extending the curve so it is asymptotic to the vertical asymptote, x=–2, and the horizontal asymptote y=2. Note that on this half-plane, since the curve cannot cross the x-axis again, there is only one way to approach the vertical asymptote: down (upward from (0,–1/2) would mean crossing the x-axis).
Question: Does the curve intersect the horizontal asymptote y=1?
Answer: Suppose there is a value of x for which y=1. "If" there truly is one, we can find it by solving the equation
Multiplying both sides by x+2 we get
Subtracting x from both sides, we get
This is impossible. That is, the equation cannot be solved for x; so the curve cannot intersect the horizontal asymptote.
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