The asymptotes are $x=\pm1$ and $y=2$. The curve intersects the horizontal asymptote at $x=7$. It intersects with the y-axis at $y=-5$. It has two turning points at $x \approx 0.1$ and $13.9$. Here is the sketch I made. However, plotting the graph with a graph plotter does not show the turning point on the right-most branch. Is this because I haven't 'zoomed enough', or is it something else?
2026-04-06 14:40:25.1775486425
Why is this an incorrect sketch for the curve $(2x^2 - x +5 ) /( x^2 -1 )$
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Putting this into Wolfy, it looks like your sketch is correct.
That min on the right is at $7+4\sqrt{3} \approx 13.928 $, with a value of $2\sqrt{3}-\frac32 \approx 1.9641 $ so you may not have gone out far enough.