Final answer:
The graph of the rational function f(x) = (4x - 3)/(x + 2) appears in the first and second quadrants with a horizontal asymptote at y = 4, and a vertical asymptote at x = -2.
Step-by-step explanation:
For the rational function f(x) = \frac{4x - 3}{x + 2}, we can determine in which quadrants the graph appears by examining the signs of the numerator and denominator. The vertical asymptote is found where the denominator is equal to zero, so x = -2 is the vertical asymptote because the function is undefined at this x-value. The horizontal asymptote is determined by the leading coefficients of the numerator and denominator, which in this case would give us y = 4 as the horizontal asymptote since the degrees of the numerator and denominator are equal and we take the ratio of the leading coefficients.
Now let us consider the quadrants. Since the horizontal asymptote is at y = 4, the graph will approach this line as x approaches infinity in either direction. When x is very large and positive, the function will approach 4 from above, so the graph is in the first quadrant. When x is very large and negative, but greater than -2, the function will approach 4 from below, so the graph is in the second quadrant. Therefore, the answer to the first question is d) First and Second quadrants, (y = 4), which was not an option given, indicating a possible error in the question.
For the second question, as mentioned, the vertical asymptote is x = -2, which is the imaginary vertical line the graph approaches but never crosses, hence the correct answer is b) (x = -2).