Graph each rational function. ƒ(x)=(2x+1)/(x²+6x+8)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{2x+1}{x^2 + 6x + 8}\).
2. Factor the denominator to find the domain restrictions and vertical asymptotes. Factor \(x^2 + 6x + 8\) as \((x + 2)(x + 4)\).
3. Determine the vertical asymptotes by setting the denominator equal to zero: solve \((x + 2)(x + 4) = 0\), which gives \(x = -2\) and \(x = -4\).
4. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the degree of the denominator (2) is greater than the numerator (1), the horizontal asymptote is \(y = 0\).
5. Find the x-intercepts by setting the numerator equal to zero: solve \(2x + 1 = 0\) to get \(x = -\frac{1}{2}\). Find the y-intercept by evaluating \(f(0) = \frac{2(0) + 1}{0^2 + 6(0) + 8} = \frac{1}{8}\).

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the behavior of rational functions involves analyzing their domain, intercepts, and asymptotes, which influence the graph's shape.

Domain and Vertical Asymptotes

The domain of a rational function excludes values that make the denominator zero. These values often correspond to vertical asymptotes, where the function approaches infinity or negative infinity, indicating points where the graph is undefined.

Horizontal and Oblique Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches infinity or negative infinity. They are determined by comparing the degrees of the numerator and denominator polynomials, guiding how the graph levels off or slopes at extremes.