Question

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

Accepted Answer

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. 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. 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. 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. 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.

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

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Key Concepts

Rational Functions and Their Domains

Factoring Polynomials

Asymptotes and Graph Behavior