Answer
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Step 1:
x = 3, -2
First, let's find the critical points of the function by setting the numerator equal to zero and solving for x: Factoring the quadratic equation, we get: Setting each factor equal to zero, we find the critical points:
Step 2:
x = -1
Now, let's find the vertical asymptote by setting the denominator equal to zero and solving for x: So, the vertical asymptote is x = - 1.
Step 3:
\frac{a_2}{b_1} = \frac{1}{1} = 1
Next, let's find the horizontal asymptote. Since the degree of the numerator is 2 and the degree of the denominator is 1, we need to divide the leading coefficients: So, the horizontal asymptote is y = 1.
Step 4:
\lim\_{x \to -1} \frac{x^2 - x - 6}{x + 1} = \lim\_{x \to -1} \frac{(x - 3)(x + 2)}{x + 1} = \frac{-4}{0}
Let's evaluate the function at the critical points and the vertical asymptote to determine the behavior of the graph near these points: When x = - 2: When x = 3: When x = - 1 (vertical asymptote): The function approaches negative infinity as x approaches - 1 from the left and positive infinity as x approaches - 1 from the right.
Step 5:
Now, let's sketch the graph using the information we've gathered: - Plot the critical points (3, 0) and (- 2, 0) - Plot the vertical asymptote (x = - 1) - Plot the horizontal asymptote (y = 1) - Sketch the graph, ensuring it approaches the vertical asymptote and horizontal asymptote appropriately
Final Answer
The graph of y = (x^2 - x - 6)/(x + 1) is sketched below. 
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