We are asked to graph the function y=(8x+3)/(2x-6) :
There is a vertical asymptote at x=3. Since the degrees of the numerator and denominator agree, the horizontal asymptote is y=4.
The y-intercept is -1/2. The x-intercept is -3/8. The graph is a hyperbola. Using long division we can rewrite as y=27/(2(x-3))+4 . Using y=1/x as the base function, the graph we want is a transformation of the graph of the base function: horizontal translation 3 units right, vertical translation 4 units up, and a vertical dilation of factor 27/2.
The graph:
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