1 identify the coordinates of the vertex and the equation of the axis of summetry of the parabola

x=2 (y-1)^2+3

A. The vertex of the parabola is (1,3), and the equation of the axis of symmetry
B. the vertex of the parabola is (3,1) and the equation of the axis of symmetry
c. the vertex of the parabola is (3,1) and the equation of the axis of symmetry is y=1
D. None of these

The equation is written in the form

x=a(y-h)^2+k
where a=2, h=1, k=3

so the vertex is at (k,h)=(3,1)

The axis of symmetry is at y=h, or y=1.

To identify the coordinates of the vertex and the equation of the axis of symmetry of the parabola, we need to rewrite the given equation in the standard form of a parabola, which is y = a(x - h)^2 + k.

Starting with the given equation x = 2(y - 1)^2 + 3:

1. First, subtract 3 from both sides to isolate the quadratic term:
x - 3 = 2(y - 1)^2

2. Divide both sides by 2 to get the coefficient of the quadratic term to be 1:
(x - 3)/2 = (y - 1)^2

Comparing this equation to the standard form, we can see that h = 3/2 and k = 1. Therefore, the vertex of the parabola is (h, k) = (3/2, 1).

The equation of the axis of symmetry for any parabola can be found by setting x - h = 0, where h is the x-coordinate of the vertex. In this case, the equation of the axis of symmetry is x - (3/2) = 0.

Simplifying this equation gives us x = 3/2, which is a vertical line passing through the x-coordinate of the vertex.

Therefore, the correct answer is option C: the vertex of the parabola is (3/2, 1), and the equation of the axis of symmetry is y = 1.