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The axis of symmetry for a parabola is a vertical line that divides the parabola into two mirror-image halves. For parabolas of the form \[ x = a(y - k)^2 + h \], the axis of symmetry is given by the equation \[ y = k \]. This line passes through the vertex and ensures that both sides of the parabola are symmetrical. In our specific example, the equation given is \[ x = 2(y - 1)^2 + 3 \]. Here, the vertex \[ (h, k) \] is \[ (3, 1) \], meaning the axis of symmetry is \[ y = 1 \]. This means the parabola is symmetrical about the line where the y-coordinate is always 1.

The focus and directrix are two important properties of a parabola. The focus lies inside the parabola, while the directrix is a line outside of it. Points on the parabola are such that the distance to the focus equals the distance to the directrix. For a horizontal parabola such as \[ x = 2 (y - 1)^2 + 3 \], the coordinates of the focus are given by: \[ \left( h + \frac{1}{4a}, k \right) \], where \[ a = 2 \]. Substituting the given values: \[ h + \frac{1}{8} = 3 + \frac{1}{8} = 3.125 \]. Hence, the focus is \[ (3.125, 1) \].

The directrix for the horizontal parabola can be found using: \[ x = h - \frac{1}{4a} \]. Substituting the values, we get: \[ x = 3 - \frac{1}{8} = 2.875 \]. Therefore, the directrix is \[ x = 2.875 \]. These two features help in sketching the parabola accurately.

The intercepts are where the parabola crosses the x-axis and y-axis. For the x-intercept, set \[ y = 0 \] and solve for \[ x \]: \[ x = 2(0 - 1)^2 + 3 \], simplifies to: \[ x = 2(1) + 3 = 5 \]. Therefore, the x-intercept is \[ (5, 0) \].

To find the y-intercept, set \[ x = 0 \] and solve for \[ y \]: \[ 0 = 2(y - 1)^2 + 3 \]. This rearranges into: \[ -3 = 2(y - 1)^2 \] \[ -\frac{3}{2} = (y - 1)^2 \]. Since a square is always non-negative and the left side is negative, the equation has no real solutions. This means the parabola does not intersect the y-axis and has no y-intercept.