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Chapter 3: Problem 114

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?

Short Answer

Expert verified
To determine if a rational function has a slant asymptote, compare the degrees of the polynomials in the numerator and the denominator. If the degree of the numerator's polynomial is exactly one more than the degree of the denominator's, it has a slant asymptote. Its equation can be obtained by dividing the numerator by the denominator.

Step by step solution

01

Identify the Degrees of Polynomials

The first step in determining the existence of a slant asymptote is to look at the degrees of the polynomials in both the numerator and the denominator of a given rational function. This can typically be done by identifying the power of the variable \(x\), which in turn is guided by the terms present.
02

Check for any Slant Asymptotes

When the degree of the polynomial in the numerator is exactly one more than the degree of the polynomial in the denominator, a slant asymptote exists. If the degrees do not fit this criterion, there will not be a slant asymptote.
03

Find the Equation of the Slant Asymptote

Once you have confirmed the presence of a slant asymptote, you can find its equation by performing the division of the numerator polynomial by the denominator polynomial. This process will generally involve long division or synthetic division of polynomials. The quotient you obtain gives the equation of the slant asymptote.

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Most popular questions from this chapter

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has a vertical asymptote given by x=1, a slant asymptote whose equation is y=x, y -intercept at 2, and x -intercepts at -1 and 2.

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Divide 737 by 21 without using a calculator. Write the answer as quotient \(+\frac{\text { remainder }}{\text { divisor }}\)

Write an equation in point-slope form and slope-intercept form of the line passing through \((-10,3)\) and \((-2,-5)\) (Section \(2.3,\) Example 3 )

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-x^{2}(x-1)(x+3)$$
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