Graph each rational function. See Examples 5–9. ƒ(x)=(x^2+2x+1)/(x^2-x-6)

Verified step by step guidance

1. Identify the vertical asymptotes by setting the denominator equal to zero and solving for \(x\).
2. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator.
3. Find the x-intercepts by setting the numerator equal to zero and solving for \(x\).
4. Calculate the y-intercept by evaluating \(f(0)\).
5. Sketch the graph using the asymptotes, intercepts, and analyzing the behavior of the function around these points.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions involves analyzing their domains, intercepts, and asymptotes, which are critical for graphing them accurately.

Finding Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. For rational functions, vertical asymptotes occur where the denominator equals zero (Q(x) = 0), while horizontal asymptotes are determined by the degrees of the polynomials in the numerator and denominator. Identifying these asymptotes is essential for sketching the graph and understanding its behavior at extreme values.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. This is particularly important for rational functions, as it helps simplify the function and identify critical points such as zeros and asymptotes. For example, the function f(x) = (x^2 + 2x + 1)/(x^2 - x - 6) can be factored to facilitate graphing.