The parabola in the form y = (x - h)^2 + k has a vertex of (h,k). The problem we have is y-3 = (x-1)^2, so we have to get the -3 from the left side and move to the right, so add 3 to both sides of the equation. That gives us y = (x-1)^2 + 3, so the vertex is (1,3)

Since the x^2 term is positive, the parabola opens up away from the x-axis, so there are no x-intercepts. If you are unsure of this you can put 0 in for y and solve for x, like i demonstrated and by the quadratic formula you will see there are no real number solutions for x, so no x-intercepts.

For the y-intercept, put 0 in for x and you'll see that y = 4.
The axis of symmetry is simply the line the goes through the vertex, cutting the parabola in half. The domain is all the possible x values, and since there are no restrictions on x, it's all real numbers. The range is all the possible y-values that the function takes on. As you can see on the graph, the lowest y value is at (1,3) so the range is from 3 to infinity.