Final answer:
To sketch the graph of (x-2)/(x²-3x-4), we find vertical and horizontal asymptotes, critical points, and points of concavity using the first and second derivatives. We then plot these features along with intercepts to create a detailed curve that captures the function's behavior.
Step-by-step explanation:
To sketch the graph of the function (x-2)/(x²-3x-4), we apply the curve sketching algorithm. First, let's find the vertical and horizontal asymptotes by factoring the denominator and setting it to zero and by evaluating the limit of the function as x approaches infinity, respectively. The denominator factors to (x-4)(x+1), leading to potential vertical asymptotes at x=4 and x=-1.
Next, we find the derivative of the function to determine critical points and analyze increasing or decreasing intervals. Similarly, the second derivative will help us find points of concavity and any points of inflection. After calculating these, we can determine the maximum and minimum values by evaluating the function at critical points and end behaviors. We also look for intercepts by setting both the numerator and denominator to zero separately.
Finally, after plotting the asymptotes, intercepts, maxima, and minima, concave up and down intervals, and inflection points, we sketch the curve paying attention to the behavior near the asymptotes and the overall shape indicated by the derivatives. This comprehensive analysis will give us a detailed graph of the given function.