Answer
$(y-1)^2-x^2=1$
Work Step by Step
Step 1. Based on the shape and orientation of the curve, we can assume the equation as $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$
Step 2. Identify the center as $(0,1)$ from the figure given in the Exercise, thus $h=0, k=1$
Step 3. Identify the vertices as $(0,2)$ and $(0,0)$, so we have $2a=2$ and $a=1$
Step 4. With the given asymptote: $y=x+1$, we have $\frac{a}{b}=1$ and $b=a=1$
Step 5. Conclusion: the equation for the given curve is $(y-1)^2-x^2=1$
