Graph each rational function. ƒ(x)=(x²+8x+16)/(x²+4x-5)

Verified step by step guidance

1. Start by factoring both the numerator and the denominator of the rational function \(f(x) = \frac{x^2 + 8x + 16}{x^2 + 4x - 5}\). For the numerator, look for two numbers that multiply to 16 and add to 8. For the denominator, find two numbers that multiply to -5 and add to 4.
2. Once factored, write the function in its simplified form, if possible, by canceling any common factors between the numerator and denominator. This will help identify any holes in the graph.
3. Determine the domain of the function by setting the denominator equal to zero and solving for \(x\). These values are excluded from the domain and correspond to vertical asymptotes or holes.
4. Find the vertical asymptotes by identifying the values of \(x\) that make the denominator zero (after simplification) but do not cancel with the numerator. Also, find the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials.
5. Calculate the \(x\)-intercepts by setting the numerator equal to zero and solving for \(x\), and find the \(y\)-intercept by evaluating \(f(0)\). Use all this information to sketch the graph, showing intercepts, asymptotes, and behavior near the asymptotes.

Key Concepts

Rational Functions and Their Domains

A rational function is a ratio of two polynomials. Understanding its domain involves identifying values that make the denominator zero, as these points are excluded and often correspond to vertical asymptotes or holes in the graph.

Factoring Polynomials

Factoring the numerator and denominator polynomials helps simplify the rational function and reveals common factors. This process is essential to identify holes and to rewrite the function in a form that makes graphing easier.

Asymptotes and Graph Behavior

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero, while horizontal or oblique asymptotes describe end behavior. Recognizing these asymptotes guides the sketching of the graph and understanding its limits.