Use completing the square before drawing a rational graph

$f(x)=x+(b-8)/4+[c-2(b-8)]/[4(x+2)]$. The oblique asymptote is $y=x+(b-8)/4$. Hence $x-5=x+(b-8)/4$, so b=−12.

For c>9, $4x^2-12x+c=4(x^2-3x)+c=4[(x-3/2)^2-9/4]+c=4(x-3/2)^2-9+c>0$ for every real x. Therefore the numerator never vanishes and C does not cut the x-axis; x=−2 is excluded from its domain.

$f(x)=x-5+53/[4(x+2)]$. Asymptotes: x=−2 and y=x−5. The y-intercept is (0,13/8). Stationary points are $(-2-\sqrt{53}/2,-7-\sqrt{53})$, a maximum≈(−5.64,−14.3), and $(-2+\sqrt{53}/2,-7+\sqrt{53})$, a minimum≈(1.64,0.280). These retain the source rounded labels. The source also marks (5,0) and (0,−5) on the oblique asymptote: these belong to the line, not the curve.