Answer
$(x-2)^2-\frac{y^2}{3}=1$
Work Step by Step
Step 1. Based on the graph given in the Exercise, we can identify the graph as a hyperbola with a general equation of $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$
Step 2. The graph gives vertices $(1,0)$ and $(3,0)$, thus we have $2a=3-1=2$ or $a=1$, and the center is the midpoint between the vertices, $C(2,0)$ which means $h=2, k=0$
Step 3. It is also given on focus at $F(4,0)$, its distance to the center gives $c=4-2=2$
Step 4. Use the relation $b^2=c^2-a^2$, we get $b^2=4-1=3$
Step 5. Conclusion: the equation for the hyperbola is $\frac{(x-2)^2}{1}-\frac{(y)^2}{3}=1$ or $(x-2)^2-\frac{y^2}{3}=1$
