Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+x−6)/(x−3)

Accepted Answer

Identify the given rational function: f(x)=x2+x−6x−3. We want to find the slant (oblique) asymptote, which occurs when the degree of the numerator is exactly one more than the degree of the denominator. Perform polynomial long division of the numerator $$x^2 + x - 6 by $$the denominator x - 3. This will help us express f(x) as a quotient plus a remainder over the divisor, which reveals the slant asymptote. Set up the division: divide the leading term of the numerator $$x^2 by $$the leading term of the denominator x to get the first term of the quotient. Then multiply the entire divisor by this term and subtract from the numerator. Repeat this process until the degree of the remainder is less than the degree of the divisor. The quotient obtained from the division (ignoring the remainder) represents the equation of the slant asymptote. Write this quotient as a linear function y = mx + b where m and b come from the division result. For graphing, use the seven-step strategy: (1) find the domain, (2) find intercepts, (3) find vertical asymptotes (where denominator is zero), (4) find the slant asymptote from the division, (5) analyze end behavior using the slant asymptote, (6) plot points around the vertical asymptote and near the slant asymptote, and (7) sketch the graph showing the behavior near asymptotes and intercepts.

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x²+x−6)/(x−3)

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Concetti chiave

Rational Functions

Slant (Oblique) Asymptotes

Polynomial Division