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)=(x3+1)/(x2+2x)

Accepted Answer

Step 1: Identify the degrees of the numerator and denominator. The numerator is $$x^{3} + 1$ (degree 3) and the denominator is x$$^{2}$$ + 2x ($$degree 2). Since the degree of the numerator is exactly one more than the degree of the denominator, a slant (oblique) asymptote exists. Step 2: Perform polynomial long division to divide the numerator $$x^{3} + 1$ by the denominator x$$^{2}$$ + 2x. $$This will give you a quotient (which represents the slant asymptote) and a remainder. Step 3: Write the quotient from the division as the equation of the slant asymptote. The slant asymptote will be of the form $$y =  ($$quotient polynomial). Step 4: Use the seven-step strategy to analyze the function: (1) Find the domain by identifying values that make the denominator zero, (2) Find intercepts by setting numerator and denominator equal to zero appropriately, (3) Determine vertical asymptotes from the denominator zeros, (4) Find the slant asymptote from the division, (5) Analyze end behavior using the slant asymptote, (6) Plot key points and asymptotes, (7) Sketch the graph using all gathered information. Step 5: Use the slant asymptote as a guide for the end behavior of the graph, noting that as $$x$$ approaches infinity or negative infinity, the graph of $$f(x)$$ will approach the slant asymptote line.

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)=(x³+1)/(x²+2x)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Slant (Oblique) Asymptotes

Polynomial Long Division

Graphing Rational Functions Using Asymptotes