Answer
$$\frac{(x-1)^2}{1}-\frac{y^2}{1^2}=1$$
Work Step by Step
The center of the hyperbola is at $(1,0)$
$$\frac{(x-1)^2}{a^2}-\frac{y^2}{b^2}=1$$
The x intercepts are 2 apart and thus 1 from the center.
$$\frac{(x-1)^2}{1}-\frac{y^2}{b^2}=1$$
Since $y=x-1$ is the slant asymptote, $b=1$ as well.
$$\frac{(x-1)^2}{1}-\frac{y^2}{1^2}=1$$
