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: