In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x²+x−6)

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Identify the domain of the function by finding the values of \( x \) that make the denominator zero. Solve \( x^2 + x - 6 = 0 \) by factoring or using the quadratic formula.

Find the vertical asymptotes by setting the denominator equal to zero and excluding any values that also make the numerator zero (which would indicate a hole instead).

Determine the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator polynomials. Since the numerator is degree 1 and the denominator is degree 2, the horizontal asymptote is \( y = 0 \).

Find the x-intercepts by setting the numerator equal to zero and solving for \( x \). For \( f(x) = \frac{x+2}{x^2 + x - 6} \), set \( x + 2 = 0 \) and solve.

Find the y-intercept by evaluating \( f(0) \), which is \( \frac{0 + 2}{0^2 + 0 - 6} \). This gives the point where the graph crosses the y-axis.

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주요 개념

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Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential because values that make Q(x) zero are excluded and often correspond to vertical asymptotes or holes in the graph.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero and the numerator is nonzero, while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.

Graphing Steps for Rational Functions

Graphing rational functions involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes. Plotting key points and analyzing limits help sketch an accurate graph, following a systematic approach such as the seven-step method.

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x2+x−6)