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

Verified step by step guidance

1. Start by factoring both the numerator and the denominator of the rational function \(f(x) = \frac{x^2 + 2x + 1}{x^2 - x - 6}\). The numerator factors as \((x + 1)^2\) and the denominator factors as \((x - 3)(x + 2)\).
2. Identify the domain restrictions by setting the denominator equal to zero and solving for \(x\): solve \((x - 3)(x + 2) = 0\). These values are excluded from the domain because they make the denominator zero, causing vertical asymptotes.
3. Determine the vertical asymptotes by using the domain restrictions found in the previous step. The vertical asymptotes occur at \(x = 3\) and \(x = -2\).
4. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are degree 2 polynomials, the horizontal asymptote is the ratio of the leading coefficients, which are both 1, so \(y = 1\).
5. Find the x-intercepts by setting the numerator equal to zero and solving for \(x\): solve \((x + 1)^2 = 0\). The y-intercept is found by evaluating \(f(0)\). Use this information along with the asymptotes to sketch the graph.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, zeros, and behavior depends on analyzing both numerator and denominator polynomials. The function is undefined where the denominator equals zero, leading to vertical asymptotes or holes.

Factoring Polynomials

Factoring involves rewriting polynomials as products of simpler polynomials. For example, x^2 + 2x + 1 factors to (x + 1)^2, and x^2 - x - 6 factors to (x - 3)(x + 2). Factoring helps identify zeros of numerator and denominator, which are critical for finding intercepts and asymptotes.

Asymptotes and Graph Behavior

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero, indicating values excluded from the domain. Holes appear if numerator and denominator share a factor. Horizontal or oblique asymptotes describe end behavior, determined by comparing degrees of numerator and denominator polynomials.